Definitions:
Individual = anyone or anything considered as though incapable of being subclassified into anything more specific than himself, herself, or itself (i.e., whatever belongs to it and is more specifical than itself is being left out of consideration).
True individual = that which is truly incapable of being subclassified into anything more specific than himself, herself, or itself.
Species = that to which an individual thing or individual person belongs if and only if he/she/it has certain well-defined characteristics that all things or persons belonging to it and only they have in common.
Subspecies of a given species = another species every individual belonging to the which, also belongs to the given species.
Disjoint species = two species such that no individual belonging to any one of these two species can belong to the other.
Population of a species = total unbroken whole number of distinct, unbroken, undivided, whole individuals belonging to the species.
Member of a given species = anything which is treated as if it had been a single, distinct, unbroken, undivided, whole individual (i.e., whatever belongs to it and is more specifical than itself is being left out of consideration), and which belongs to the given species.
Sequence = a species all of whose members are arranged in some particular order of succession, and which is considered with regard to the particular order of succession in which all of its members are arranged.
Ordinal number = any one of those members of that sequence in which all the members are specifically designed in order to show order of succession, and immediately after each such member, there is one and only one member, being the nearest possible posterior member, such that the two differ from each other by exactly that number called by the name of "one", and between these two there cannot exist a third, and all of the respective differences between the members and their respective nearest possible posterior members are exactly equal to one another.
Successor ordinal = any ordinal number which immediately follows some previous ordinal number.
Asymptotic ordinal = any ordinal which is neither the first of all the ordinals, nor a successor ordinal.
Finite number = a number which, if considered as ordinal, comes before every conceivable asymptotic ordinal.
Infinite number = Transfinite number = a number which, if considered as ordinal, comes immediately or mediately after every conceivable finite ordinal, and is either itself an asymptotic ordinal or comes (mediately or immediately) after an asymptotic ordinal.
Set = any one of those differing species, all belonging to a common kind, for the whole of which, there is some formal axiomatic system (even if we have not yet fully formulated it!), according to which, in each such species belonging to this one kind, the assumption that the species is per se a completed actual distinct unbroken unified determinate single whole, bounded on all sides on which it is populated (even if that population is infinite in size), and the size of which is merely the total unbroken whole number of distinct, unbroken, whole, and undivided individuals belonging to the species, cannot, within just that formal axiomatic system, to be proven to be contrary to the definitions, axioms, and postulates on which that formal axiomatic system is founded, nor to any one or more of the theorems which follow irrefutably from just those very definitions, axioms, and postulates, even if there are always other members of the species than any specified but not actually infinite number (however great) of individual members of that species.
Element of a set = any individual belonging to that set.
Empty Set = a species to which nothing nor anyone belongs and is entirely unpopulated.
Axioms or Postulates
1. In every valid system of classification, the principles of classification employed ought to be consistent and unique.
2. In every valid system of classification, any two differing classes or species which happen to be at the same exact level of specificity ought to be mutually exclusive: each individual must fit into only one such of those two classes, preventing ambiguity or overlap between groups.
3. In every valid system of classification, every individual case, even if not yet known, ought to be capable of being individually absorbed, provided that enough effort and mental discipline is applied in making the necessary amount of scrutiny and inquiry, even if none of the other cases ever manage to get absorbed.
4. No true individual is a species; and conversely, every individual—whether truly individual or only treated as such—is at least one degree more specific than every species, however specific, to which it belongs.
5. In every coherent and well-defined characteristic or attribute, all of the individuals possessing that characteristic or attribute in common are alike members of one and only one unique species to which everything or everyone belongs if and only if he, she, or it possesses that characteristic or attribute.
6. Unto every given species of differing and mutually exclusive subspecies, even if they are not all at the same level of specificity, there is yet another species containing exactly one member of each such subspecies of the given species, but no other members except members of the such subspecies of the given species.
Objective or Goal: We desire to formulate some formal axiomatic theory of sets, which agrees with these six above axioms or postulates. Note: Axiom #6 can be written in terms of propositions:
In every given multitude of mutually exclusive given propositions each of which concerns its own distinct variable individual, but one and only such variable individual, there is a proposition which, for each of the given propositions in the given multitude, is true for exactly one value of the only variable concerned by the given proposition, for which such a given proposition in the given multitude is true, even if it is not yet known how to formulate or write that proposition unambiguously in writing, speech, prose, or poetry.
And, Deo Volente, I will prove that the theory known as ZFC set theory (Zermelo-Frankel set theory with the Axiom of Choice) is the only formal axiomatical theory of sets which can meet our assigned objective or goal.
Fundamental Theses of Set Theory
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StudentDriver
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Re: Fundamental Theses of Set Theory
Why you made this set theory rather than following the conventional naive set theory route?
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Casus_Conscientiae
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Re: Fundamental Theses of Set Theory
That is really an excellent and fundamental question !StudentDriver wrote: Fri Jun 12, 2026 6:43 pm Why you made this set theory rather than following the conventional naive set theory route?
My answer is:
Because first of all, each of the axioms of a true set theory are contingent on something more fundamental than itself: namely the foundational principles of a valid system of classification.
Second of all, conventional naive set theory is fundamentally flawed and blemished, not by accident, but by virtue of its intrinsic and inherent nature and constitution. By allowing any arbitrary property to define a set (Unrestricted Comprehension), naive set theory contains the seeds of its own destruction via logical paradoxes. My "Fundamental Theses" are designed to construct a rigorous bridge between classical Aristotelian logic (classification by genus and species) and modern axiomatic set theory (ZFC) precisely to avoid the traps of the naive approach. Here is exactly why this framework deviates from the naive route:
It is a tenet of naive set theory that every conceivable species is ipso facto respectively a set, and that the species to which everything belongs if and only if it has certain well-defined properties common to all members and only members of that species is ipso facto a set. Because no individual is a species, a set cannot be one of its own elements: the set is at least one step less specific than any one of its own elements! Therefore that set must belong to the species S of all sets that don't contain themselves as one of their own individual elements!
According to naive set theory, S is a set. Either it contains itself as one of its own individual members or it doesn't. If it does, well then by the definition of S, the set S cannot contain itself as one of its own individual elements!
But if it doesn't, well then by the definition of S, it too is one of the individual members of S itself: therefore it contains itself as one of its own individual members. Each of which is impossible.
This paradox is called the Russell Paradox.
"The lover of all those - and only those - who don't love themselves"? Call him L. Now when we say "the lover of all those - and only those - who don't love themselves", we are talking about somebody who loves all persons other than himself if and only they don't love themselves. If this condition "other than himself" is overlooked, well then consider L = the lover of all those - and only those - who don't love themselves, but then we can pose the question - does L love himself? If he does well then by definition he can't love himself. But if he doesn't, well then by definition he has to love himself: both of which are utterly impossible. This simple condition consisting of the phrase "other than himself" is essential in order to preserve logical coherence.
But naive set theory predicts that there is a set to which everything belongs if and only if it is a set that isn't one of its own individual members. But this automatically results in a logical contradiction. Because naive set theory cannot resolve this contradiction, the entire system collapses into triviality via the principle of explosion (where any statement can be proven true).
My definition of a Set explicitly anticipates and neutralizes this by stating that a species is only a set if the assumption of it being a "completed actual distinct unbroken unified determinate single whole... cannot, within just that formal axiomatic system, be proven to be contrary to the definitions, axioms, and postulates." This introduces a criterion of provable consistency that naive set theory lacks.
Third of all, You'll notice that Axiom 5 sounds superficially like naive comprehension: "In every coherent and well-defined characteristic... all of the individuals possessing that characteristic... are alike members of one and only one unique species." However, notice the crucial ontological distinction in my framework: A property defines a Species. A Species is not automatically a Set. A species only graduates to a "Set" if it can be consistently bounded and treated as a completed mathematical object within an axiomatic framework. This mirrors the transition from Cantor's naive sets to Zermelo's Axiom of Separation ($x \in A \text{ such that } \phi(x)$), where we must already possess a valid "domain" before we can filter it by a property. Furthermore, Axiom 6 (and its propositional variant) is a deliberate, foundational phrasing of the Axiom of Choice (AC). Naive set theory took choice for granted, which led to deep conceptual confusion when infinite collections were involved. By explicitly isolating this as a postulate concerning mutually exclusive subspecies, I am preparing the ground to prove that ZFC is the natural mathematical realization of these intuitive rules of classification.
Fourthly: Naive set theory divorced itself from traditional logic and structural classification. By using terms like Individual, Species, Subspecies, and Level of Specificity, this system roots mathematical sets in the rigorous ground of formal classification systems. Axioms 1, 2, and 3 ensure that our system of classification is consistent, partition-based (mutually exclusive at the same level of specificity), and comprehensive. This provides an epistemological framework for why we do set theory, rather than just throwing objects into abstract collections without a underlying philosophy of identity.
Conclusion: I didn't choose the naive route because the naive route leads to mathematical lawlessness.
Instead, these Theses formalize the rigorous laws of classification (Axioms 1–4), bridge them to the concept of properties (Axiom 5), introduce the necessity of selection (Axiom 6), and restrict the definition of a Set to those species that do not trigger logical contradictions. This is the exact philosophical scaffolding required to demonstrate why ZFC isn't just an arbitrary collection of rules, but the inevitable destination of consistent thought.
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StudentDriver
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Re: Fundamental Theses of Set Theory
hold on a min, what's wrong with zfc? You trying to eliminate the Banach-Tarski paradox or somethin? I know zfc avoids Russel paradox. I know ZF avoids Banach-Tarski. In my opinion I dont mind ZFC. So why not use ZF instead?
And btw Great ideas man. Can I ask you, why is it that you pursue to change zfc completely? were you not satisfy with it? Did the Banach Tarski bother you that much??
And btw Great ideas man. Can I ask you, why is it that you pursue to change zfc completely? were you not satisfy with it? Did the Banach Tarski bother you that much??
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Casus_Conscientiae
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Re: Fundamental Theses of Set Theory
Thesis 0: Recapitulation of Axiom #4. No true individual is a species; and conversely, every individual—whether truly individual or only treated as such—is at least one degree more specific than every species, however specific, to which it belongs. Therefore, no species is one of its own individual members.
Thesis 1: Principle of Extensionality. It is a self-evident truth that any two given species are identical if and only if every individual member belonging to the first of the two species ipso facto belongs to the second and every individual member belonging to the second ipso facto belongs to the first.
Thesis 2: Recapitulation of Axiom #5. Principle of Unique Speciation: In every coherent and well-defined characteristic or attribute, all of the individuals possessing that characteristic or attribute in common are alike members of one and only one unique species to which everything or everyone belongs if and only if he, she, or it possesses that characteristic or attribute.
Thesis 3: There is no given species to which at least one individual (member) belongs, and yet to which, also belongs any individual (member) whose common subspecies with the given species also has at least one individual (member) belonging to this common subspecies. Proof: Or else, suppose that there had been some species S (to which there belongs at least one individual) containing some individual (member) e such that T, which is the most general possible common subspecies belonging to both e and S, has at least one individual or item belonging to T. But then if e is a true individual, well then e cannot have more than one individual, therefore neither can T have more than one individual. But if e is not a true individual, well then whatever individuals more specific than itself it might have, yet even so, they do not need to be considered, since e is being treated as if it had been a true individual: since whatever belongs to e and is more specifical than e has by definition, been left out of consideration. So in either case, of all the entities none of which have been left out of consideration, T cannot have more than one such individual! Yet T is a subspecies of e. Now either e is a true individual or it is merely an item. If e is a true individual, well then: ⦁ Either T and e are identical, which means that e is both a true individual and a species, which is impossible, ⦁ or else T and e are not identical, meaning e itself has been subclassified into two different subspecies, the first being T, and the second being a different subspecies U which does not have any individual in common with T: but then even T itself is more specifical than e itself, which also is impossible because e is an individual, and no species can be more specifical than any one given individual. But if e is not a true individual, well then by definition e is being treated as if it had been an individual, since whatever belongs to e and is more specifical than e has by definition, been left out of consideration; and ⦁ Either T and e are identical, which means that e is being treated as if it had been both an individual and a species; yet this means that T and e are at the same level of specificity: which is impossible, because no item can belong to T itself without also belonging to both e and T, and therefore more specific than e itself, which is impossible, because whatever belongs to e and is more specifical than e has by definition, been left out of consideration; ⦁ or else T and e are not identical, meaning e itself has been subclassified into two different subspecies, the first being T, and the second being a different subspecies U which does not have any individual in common with T: but then even T itself is more specifical than e itself, which also is impossible because nothing other than e itself can belong to e without being more specifical than e: and yet whatever belongs to e and is more specifical than e has by definition, been left out of consideration!! Thus in either case whether e is a true individual or not, the assumed species S cannot exist. **QED
Corollary I: Principle of Regularity. Every given species to which belongs at least one individual member, has among all the individuals belong to the given species itself, at least one which is disjoint from the given species.
Corollary II: No theory of sets in which the axiom of regularity is denied or rejected can accomplish the desired goals for our formal axiomatic theory of sets.
Thesis 4: Principle of Foundation. An unterminating (or even unboundedly prolongable) sequence of species (each with at least one individual member belonging to it) such that each such species is an individual member belong to the immediately previous species in the sequence cannot exist. Proof: Or else suppose that in the unterminating sequence X_1, X_2, ..., X_n, ...: the items in the sequence could be in such a relationship according to the which: X_1 ∋ X_2 ∋ ... ∋ X_n ∋ .... But then the sequence itself is already a species S with at least 1 individual member belonging to it! According to the Corollary of Thesis 3, some individual member X_k of S is disjoint from S: therefore in X_k, no individual (whether truly an individual or merely being treated as an individual) belonging to X_k belongs to the species S. Yet X_{k+1} ∈ X_k ∈ S: therefore X_k, S have a common subspecies to which belongs at least one individual member - which is impossible!!! Therefore the proposed infinite sequence cannot exist. QED
Thesis 1: Principle of Extensionality. It is a self-evident truth that any two given species are identical if and only if every individual member belonging to the first of the two species ipso facto belongs to the second and every individual member belonging to the second ipso facto belongs to the first.
Thesis 2: Recapitulation of Axiom #5. Principle of Unique Speciation: In every coherent and well-defined characteristic or attribute, all of the individuals possessing that characteristic or attribute in common are alike members of one and only one unique species to which everything or everyone belongs if and only if he, she, or it possesses that characteristic or attribute.
Thesis 3: There is no given species to which at least one individual (member) belongs, and yet to which, also belongs any individual (member) whose common subspecies with the given species also has at least one individual (member) belonging to this common subspecies. Proof: Or else, suppose that there had been some species S (to which there belongs at least one individual) containing some individual (member) e such that T, which is the most general possible common subspecies belonging to both e and S, has at least one individual or item belonging to T. But then if e is a true individual, well then e cannot have more than one individual, therefore neither can T have more than one individual. But if e is not a true individual, well then whatever individuals more specific than itself it might have, yet even so, they do not need to be considered, since e is being treated as if it had been a true individual: since whatever belongs to e and is more specifical than e has by definition, been left out of consideration. So in either case, of all the entities none of which have been left out of consideration, T cannot have more than one such individual! Yet T is a subspecies of e. Now either e is a true individual or it is merely an item. If e is a true individual, well then: ⦁ Either T and e are identical, which means that e is both a true individual and a species, which is impossible, ⦁ or else T and e are not identical, meaning e itself has been subclassified into two different subspecies, the first being T, and the second being a different subspecies U which does not have any individual in common with T: but then even T itself is more specifical than e itself, which also is impossible because e is an individual, and no species can be more specifical than any one given individual. But if e is not a true individual, well then by definition e is being treated as if it had been an individual, since whatever belongs to e and is more specifical than e has by definition, been left out of consideration; and ⦁ Either T and e are identical, which means that e is being treated as if it had been both an individual and a species; yet this means that T and e are at the same level of specificity: which is impossible, because no item can belong to T itself without also belonging to both e and T, and therefore more specific than e itself, which is impossible, because whatever belongs to e and is more specifical than e has by definition, been left out of consideration; ⦁ or else T and e are not identical, meaning e itself has been subclassified into two different subspecies, the first being T, and the second being a different subspecies U which does not have any individual in common with T: but then even T itself is more specifical than e itself, which also is impossible because nothing other than e itself can belong to e without being more specifical than e: and yet whatever belongs to e and is more specifical than e has by definition, been left out of consideration!! Thus in either case whether e is a true individual or not, the assumed species S cannot exist. **QED
Corollary I: Principle of Regularity. Every given species to which belongs at least one individual member, has among all the individuals belong to the given species itself, at least one which is disjoint from the given species.
Corollary II: No theory of sets in which the axiom of regularity is denied or rejected can accomplish the desired goals for our formal axiomatic theory of sets.
Thesis 4: Principle of Foundation. An unterminating (or even unboundedly prolongable) sequence of species (each with at least one individual member belonging to it) such that each such species is an individual member belong to the immediately previous species in the sequence cannot exist. Proof: Or else suppose that in the unterminating sequence X_1, X_2, ..., X_n, ...: the items in the sequence could be in such a relationship according to the which: X_1 ∋ X_2 ∋ ... ∋ X_n ∋ .... But then the sequence itself is already a species S with at least 1 individual member belonging to it! According to the Corollary of Thesis 3, some individual member X_k of S is disjoint from S: therefore in X_k, no individual (whether truly an individual or merely being treated as an individual) belonging to X_k belongs to the species S. Yet X_{k+1} ∈ X_k ∈ S: therefore X_k, S have a common subspecies to which belongs at least one individual member - which is impossible!!! Therefore the proposed infinite sequence cannot exist. QED
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Casus_Conscientiae
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Re: Fundamental Theses of Set Theory
Thesis 5: Axiom of Pairing. For every two distinct individuals, it is self-evident that whatever contains both and only those two individuals as elements is also ipso facto not merely a species, but also a set. Proof: For suppose that C contains only a, b and that a and b are individuals. Then the side on which a and b are both situated is precisely the very side on which C is populated, and a and b are in fact the very bounds on C on that very side! QED
Corollary: No theory of 'sets' in which the axiom of pairing is denied can accomplish the desired goals for our formal axiomatic theory of sets.
Thesis 6: Axiom of Deflation. Every subspecies of a given species that meets the definition of a set is also ipso facto a set. Proof: For suppose that T is a subspecies of the set S. Then by definition it can be assumed without risk of logical self-contradictions, that S is bounded on all sides on which it is populated. But every individual member of T is an individual member of S, therefore by construction T itself is bounded on all sides on which it is populated. QED
Corollary: No theory of 'sets' in which the axiom of deflation is denied can accomplish the desired goals for our formal axiomatic theory of sets.
Corollary I: No species having a subspecies too big to be a set is a set.
Corollary II: Zermelo-Frankel Statement of the Axiom of Subsets. In every given property or attribute, and every given one or more of certain specified parameters, and every set and specification of the values of the specified parameters, that to which every individual belongs if and only if he, she, or it is one of those elements each of which belongs to the set and also has the given property given the specification of the values of the specified parameters, is also a set: If P is a property, attribute, proposition, hypothesis, fact, or opinion with parameters **q**, in the sense that P(x, **Q**) = true if and only if the individual x has that property P given that **q** = **Q** then for every set X and value **q**ʹ of **q**, that to which every individual belongs if and only if he, she, or it is one of those elements of X that has the property P, viz. P(x, **q**ʹ) = true, is also a set; for by construction that "species" Y of only all elements of X for which P(x, **q**ʹ) = true is a subspecies of X, and X is a set, therefore by thesis 3, Y is also a set!
Corollary III: The common section of two sets, which are not disjoint from each other, is also a set.
Definition: Subset of a given set = another set, every individual member belonging to the which, also belongs to the given set.
Thesis 7: Axiom of Union. In every given set every one of whose individual members is also a set, that species to which every individual belongs if and only if he, she, or it belongs to at least one of the sets in the given set is also a set: this latter species is called the Union of all the sets in the given set.
Proof: Obviously if the given set X exists, it is self-evident that Y, being the species to which every individual belongs if and only if he, she, or it belongs to at least one of the sets in X, also exists. But every one of the sets in X can be assumed to be bounded (on all sides on which it is populated!) without risk of logical self-contradiction. And the same is true also for X itself.
What happens if we now assume that Y is also bounded on all sides on which it is populated? We assume that every one of the members of X exists within the confines of X. Denying that Y is also likewise bounded on all sides on which it is populated is like denying that when bounded jars of pickles are in the same bounded room, the entire whole of those pickles is likewise ipso facto bounded. So assuming that Y is a completed whole bounded on all sides on which it is populated, and whose total size is merely its population yields no self-contradictions so far: therefore Y is a set. QED
Definition: Function = in any set-theoretical sense: any correspondence rule from one species to another according to which there cannot be two or more different members of the second species corresponding to the same member of the first.
Thesis 8: Axiom of Replacement. In every given function governing the individual members of a given set, the species to which every individual belongs if & only if he, she, or it is one of the possible corresponding values of the given function governing the individual members of the given set is also a set.
Proof: For we can assume that the set S is completed whole bounded on all sides on which it is populated, and whose total size is merely its population. Now we take any given function f and apply it to S to get the species T which every individual belongs if & only if he, she, or it is one of the possible corresponding values of f governing the individual members of S. But then the population of T cannot be > the population of S: but we have an indefinite species W which was originally identical to S. By means of f, the species W has been "evolved" to the species T. But the species W is bounded, and its population during the process cannot be > that of S: the number of values of f ≤ the number of members of S upon each of which f produces its respective effect ≤ the total number of members of S. This in itself is strong presumptive evidence in favor of the hypothesis that T is bounded on all sides on which it is populated.
Also, T is a completed whole: none of the actual respective effects of f upon the respective members of S are lacking. For these reasons, it follows that T is a set. QED
Corollary: No theory of "sets" in which the axiom schema of replacement is denied can accomplish the desired goals for our formal axiomatic theory of sets.
Corollary: No theory of 'sets' in which the axiom of pairing is denied can accomplish the desired goals for our formal axiomatic theory of sets.
Thesis 6: Axiom of Deflation. Every subspecies of a given species that meets the definition of a set is also ipso facto a set. Proof: For suppose that T is a subspecies of the set S. Then by definition it can be assumed without risk of logical self-contradictions, that S is bounded on all sides on which it is populated. But every individual member of T is an individual member of S, therefore by construction T itself is bounded on all sides on which it is populated. QED
Corollary: No theory of 'sets' in which the axiom of deflation is denied can accomplish the desired goals for our formal axiomatic theory of sets.
Corollary I: No species having a subspecies too big to be a set is a set.
Corollary II: Zermelo-Frankel Statement of the Axiom of Subsets. In every given property or attribute, and every given one or more of certain specified parameters, and every set and specification of the values of the specified parameters, that to which every individual belongs if and only if he, she, or it is one of those elements each of which belongs to the set and also has the given property given the specification of the values of the specified parameters, is also a set: If P is a property, attribute, proposition, hypothesis, fact, or opinion with parameters **q**, in the sense that P(x, **Q**) = true if and only if the individual x has that property P given that **q** = **Q** then for every set X and value **q**ʹ of **q**, that to which every individual belongs if and only if he, she, or it is one of those elements of X that has the property P, viz. P(x, **q**ʹ) = true, is also a set; for by construction that "species" Y of only all elements of X for which P(x, **q**ʹ) = true is a subspecies of X, and X is a set, therefore by thesis 3, Y is also a set!
Corollary III: The common section of two sets, which are not disjoint from each other, is also a set.
Definition: Subset of a given set = another set, every individual member belonging to the which, also belongs to the given set.
Thesis 7: Axiom of Union. In every given set every one of whose individual members is also a set, that species to which every individual belongs if and only if he, she, or it belongs to at least one of the sets in the given set is also a set: this latter species is called the Union of all the sets in the given set.
Proof: Obviously if the given set X exists, it is self-evident that Y, being the species to which every individual belongs if and only if he, she, or it belongs to at least one of the sets in X, also exists. But every one of the sets in X can be assumed to be bounded (on all sides on which it is populated!) without risk of logical self-contradiction. And the same is true also for X itself.
What happens if we now assume that Y is also bounded on all sides on which it is populated? We assume that every one of the members of X exists within the confines of X. Denying that Y is also likewise bounded on all sides on which it is populated is like denying that when bounded jars of pickles are in the same bounded room, the entire whole of those pickles is likewise ipso facto bounded. So assuming that Y is a completed whole bounded on all sides on which it is populated, and whose total size is merely its population yields no self-contradictions so far: therefore Y is a set. QED
Definition: Function = in any set-theoretical sense: any correspondence rule from one species to another according to which there cannot be two or more different members of the second species corresponding to the same member of the first.
Thesis 8: Axiom of Replacement. In every given function governing the individual members of a given set, the species to which every individual belongs if & only if he, she, or it is one of the possible corresponding values of the given function governing the individual members of the given set is also a set.
Proof: For we can assume that the set S is completed whole bounded on all sides on which it is populated, and whose total size is merely its population. Now we take any given function f and apply it to S to get the species T which every individual belongs if & only if he, she, or it is one of the possible corresponding values of f governing the individual members of S. But then the population of T cannot be > the population of S: but we have an indefinite species W which was originally identical to S. By means of f, the species W has been "evolved" to the species T. But the species W is bounded, and its population during the process cannot be > that of S: the number of values of f ≤ the number of members of S upon each of which f produces its respective effect ≤ the total number of members of S. This in itself is strong presumptive evidence in favor of the hypothesis that T is bounded on all sides on which it is populated.
Also, T is a completed whole: none of the actual respective effects of f upon the respective members of S are lacking. For these reasons, it follows that T is a set. QED
Corollary: No theory of "sets" in which the axiom schema of replacement is denied can accomplish the desired goals for our formal axiomatic theory of sets.
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Casus_Conscientiae
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Re: Fundamental Theses of Set Theory
The Axiom of Infinity
Thesis 9: Every given species, the first of whose individuals is any one given individual member, and which is of such a nature that for every selected individual belonging to the given species, the least common container of the selected individual and that species whose only individual member is the selected individual is also a member of the given species, is also a set. Proof: For we can do a graphical representation in which the members of W are represented by points. Then let b ∈ W be the first member of b. Also it follows that the species J.b whose only individual member is b also belongs to W. And the same is true for J⁽²⁾.b = J.J.b. Let J⁽ⁿ⁾.b = J.J⁽ⁿ⁻¹⁾.b. Then the differing J⁽ⁿ⁾.b's have different levels of specificity: the lower the number n, the greater the level of specificity: therefore no two differing J⁽ⁿ⁾.b's are the same exact individual! We will now represent J⁽ⁿ⁾.b in the following manner:
The point B represents b,
J.b is represented by the point a⁽¹⁾, which bisects the finite straight line segment AB;
For n = 2, 3, ...: a⁽ⁿ⁾ represents the point J⁽ⁿ⁾.b and Aa⁽ⁿ⁾ bisects Aa⁽ⁿ⁻¹⁾;
Then it follows that AB contains all the points a⁽ⁿ⁾, and the sequence of points B, a⁽¹⁾, ... , a⁽ⁿ⁾, ... is a graphical representation of W; but the sequence B, a⁽¹⁾, ... , a⁽ⁿ⁾, ... is fully contained in AB! And in the sequence B, a⁽¹⁾, ... , a⁽ⁿ⁾, ... in which B represents the instant at which the task begins and A represents a later instant of time, that sequence would have been a completed finished reality had it been possible first of all, to carry on a series of consecutive steps in which each step takes only half the time as the previous step, and secondly, had the whole series of consecutive steps been actually carried out; for in fact, had these two conditions been fulfilled, the whole series would have been completed in just twice the time as the first step! This in itself is presumptive evidence in favor of assuming that W is a completed reality whose size is merely its population. Therefore W is a set. QED
Corollary: Likewise it can be proven that the species to which everything belongs if and if only if it is a natural number is also a set.
Thesis 9a: No given species, the first of whose individuals is any one given individual member, and which is of such a nature that for every selected individual belonging to the given species, the least common container of the selected individual and that species whose only individual member is the selected individual is also a member of the given species has only a finite population. Proof: Or else suppose that the population of W, being a species, the first of whose individuals is b ∈ W, and which is of such a nature that for every selected individual j belonging to W, the least common container of j and that species S who only individual member is j is also a member of the given W, had had been finite.
We will call this population P, which is an unbroken whole number. Then by hypothesis b ∈ W. Also it follows that the species J.b whose only individual member is b also belongs to W. And the same is true for J⁽²⁾.b = J.J.b. Let J⁽ⁿ⁾.b = J.J⁽ⁿ⁻¹⁾.b. Then the differing J⁽ⁿ⁾.b's have different levels of specificity: the lower the number n, the greater the level of specificity: therefore no two differing J⁽ⁿ⁾.b's are the same exact individual! Then even if n > 1000 × P, J⁽ⁿ⁾.b ∈ W or suppose that there is a case in which J⁽ⁿ⁾.b does not belong to W. Then there must exist a least such n, we call it k. But then J⁽ᵏ⁻¹⁾.b ∈ W therefore J⁽ᵏ⁾.b ∈ W, contrary to the hypothesis! Therefore: Even if n > 1000 × P, yet even so, J⁽ⁿ⁾.b ∈ W Yet all of the J⁽ⁿ⁾.b's are distinct, therefore W would have have a population of over 1000 × P, contrary to the hypothesis! Therefore the population of W is not finite, but contains subspecies with as great a population as one desires! QED
Corollary I: Such a set is truly infinite.
Corollary II: No theory of "sets" in which the axiom of infinity is rejected can accomplish the desired goals for our formal axiomatic theory of sets.
Thesis 9: Every given species, the first of whose individuals is any one given individual member, and which is of such a nature that for every selected individual belonging to the given species, the least common container of the selected individual and that species whose only individual member is the selected individual is also a member of the given species, is also a set. Proof: For we can do a graphical representation in which the members of W are represented by points. Then let b ∈ W be the first member of b. Also it follows that the species J.b whose only individual member is b also belongs to W. And the same is true for J⁽²⁾.b = J.J.b. Let J⁽ⁿ⁾.b = J.J⁽ⁿ⁻¹⁾.b. Then the differing J⁽ⁿ⁾.b's have different levels of specificity: the lower the number n, the greater the level of specificity: therefore no two differing J⁽ⁿ⁾.b's are the same exact individual! We will now represent J⁽ⁿ⁾.b in the following manner:
The point B represents b,
J.b is represented by the point a⁽¹⁾, which bisects the finite straight line segment AB;
For n = 2, 3, ...: a⁽ⁿ⁾ represents the point J⁽ⁿ⁾.b and Aa⁽ⁿ⁾ bisects Aa⁽ⁿ⁻¹⁾;
Then it follows that AB contains all the points a⁽ⁿ⁾, and the sequence of points B, a⁽¹⁾, ... , a⁽ⁿ⁾, ... is a graphical representation of W; but the sequence B, a⁽¹⁾, ... , a⁽ⁿ⁾, ... is fully contained in AB! And in the sequence B, a⁽¹⁾, ... , a⁽ⁿ⁾, ... in which B represents the instant at which the task begins and A represents a later instant of time, that sequence would have been a completed finished reality had it been possible first of all, to carry on a series of consecutive steps in which each step takes only half the time as the previous step, and secondly, had the whole series of consecutive steps been actually carried out; for in fact, had these two conditions been fulfilled, the whole series would have been completed in just twice the time as the first step! This in itself is presumptive evidence in favor of assuming that W is a completed reality whose size is merely its population. Therefore W is a set. QED
Corollary: Likewise it can be proven that the species to which everything belongs if and if only if it is a natural number is also a set.
Thesis 9a: No given species, the first of whose individuals is any one given individual member, and which is of such a nature that for every selected individual belonging to the given species, the least common container of the selected individual and that species whose only individual member is the selected individual is also a member of the given species has only a finite population. Proof: Or else suppose that the population of W, being a species, the first of whose individuals is b ∈ W, and which is of such a nature that for every selected individual j belonging to W, the least common container of j and that species S who only individual member is j is also a member of the given W, had had been finite.
We will call this population P, which is an unbroken whole number. Then by hypothesis b ∈ W. Also it follows that the species J.b whose only individual member is b also belongs to W. And the same is true for J⁽²⁾.b = J.J.b. Let J⁽ⁿ⁾.b = J.J⁽ⁿ⁻¹⁾.b. Then the differing J⁽ⁿ⁾.b's have different levels of specificity: the lower the number n, the greater the level of specificity: therefore no two differing J⁽ⁿ⁾.b's are the same exact individual! Then even if n > 1000 × P, J⁽ⁿ⁾.b ∈ W or suppose that there is a case in which J⁽ⁿ⁾.b does not belong to W. Then there must exist a least such n, we call it k. But then J⁽ᵏ⁻¹⁾.b ∈ W therefore J⁽ᵏ⁾.b ∈ W, contrary to the hypothesis! Therefore: Even if n > 1000 × P, yet even so, J⁽ⁿ⁾.b ∈ W Yet all of the J⁽ⁿ⁾.b's are distinct, therefore W would have have a population of over 1000 × P, contrary to the hypothesis! Therefore the population of W is not finite, but contains subspecies with as great a population as one desires! QED
Corollary I: Such a set is truly infinite.
Corollary II: No theory of "sets" in which the axiom of infinity is rejected can accomplish the desired goals for our formal axiomatic theory of sets.
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Casus_Conscientiae
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Re: Fundamental Theses of Set Theory
Axiom of the Power Set & Axiom of Choice
Thesis 10: 10a: Every nonempty set has among its individual members, at least one which is disjoint with the set; 10b: No set is one of its own elements; 10c: An infinite sequence of non-empty sets such that each such set is an individual element of the immediately previous set in the sequence cannot exist.
Thesis 11: Axiom of the Power Set. In every given set, the species to which every individual belongs if & only if he, she, it is either the empty set, the given set itself, or one of the subsets of the given set is also a set. Such a set is called the Power Set = of a given set: the set to which every element belongs if & only if it is either the empty set, or else the given set itself, or else yet, one of the subsets of the given set. If a given set has population N, its power set has population 2^N.
Proof: Thus obviously the Axiom of the Power set is true for finite sets. The as of yet unproven part is that it is also true for infinite sets. For the infinite case, suppose that S is a set with an infinitely great population. Let W be that to which every individual belongs if & only if he, she, it is either the empty set, the given set S itself, or one of the subsets of S. No member of W can be "simpler" than the empty set, nor more "complicated" than S itself. This mere fact is ipso facto presumptive evidence in favor of the conjecture that W is bounded on all sides on which it is populated.
Also we can assume that W is a compled total reality, lacking nothing essential to its "perfection". Every contribution that an individual makes to the population size is individually discrete and, and all such contributions and equal and intrinsically indivisible provided that neither dies nor reproduces. Therefore the size of W is merely its population. The assumption that W is not a completed total reality entails that W is lacking certain individuals which do happen to meet the necessary and sufficient conditions for membership in W: which is contary to hypothesis!
For all these reasons, W is a set. QED
Corollary: No theory of 'sets' in which the axiom of the power set is denied for infinite sets can accomplish our desired objectives any more than if that axiom had been denied for finite sets.
Thesis 12: Recapitulation of Axiom #6. Axiom of Choice: Unto every given species of differing and mutually exclusive subspecies, even if they are not all at the same level of specificity, there is yet another species containing exactly one member of each such subspecies of the given species, but no other members except members of the such subspecies of the given species.
Thesis 12a: If a given species of different and mutually exclusive subspecies is a set, well then, every possible species containing exactly one member from each subspecies of the given species, but no other members except members of the subspecies of the given species, is also a set. Proof: Suppose that M is a species having different (and mutually exclusive) subspecies. Well then let Q be a species containing one and only one member from each subspecies s of M, but no other members except members of the subspecies of M. Only one member m.s of species Q corresponds to the subspecies s: any formula F for finding m.s from s is a correspondence rule from M to Q according to which two different members of Q cannot correspond to the same subspecies of M: therefore that formula is a function governing the subspecies of M. Also Q does not have more than one member of s, or else any 2 of those two members would be both in Q and both of s - contrary to hypothesis. And every member of M must be one of the members of the species in M. But the subspecies of M are mutually exclusive: meaning that no member of Q can belong to two such different subspecies in common. The same member of Q cannot correspond to two different "members" of M. But now suppose that M is a set. But then by thesis 5, the possible species R of values of m.s is a set, and different m.s implies different s, and the same m.s implies the same s. The possible values of m.s are all contained in Q, therefore every member of R is contained in Q. Also every member of Q is contained in R. Let K be a possible value of m.s. Then it is in species Q. But now let G be in Q. Then by definition it has got to be a member of one of the subspecies of M. Therefore G has to be m.t where t is a subspecies of M; therefore G is in R. Therefore every member of Q is contained in R. Therefore R and Q are exactly identical. Therefore Q itself is a set! QED
Corollary: No theory of sets in which the axiom of choice is denied can accomplish our desired objectives.
Thesis 10: 10a: Every nonempty set has among its individual members, at least one which is disjoint with the set; 10b: No set is one of its own elements; 10c: An infinite sequence of non-empty sets such that each such set is an individual element of the immediately previous set in the sequence cannot exist.
Thesis 11: Axiom of the Power Set. In every given set, the species to which every individual belongs if & only if he, she, it is either the empty set, the given set itself, or one of the subsets of the given set is also a set. Such a set is called the Power Set = of a given set: the set to which every element belongs if & only if it is either the empty set, or else the given set itself, or else yet, one of the subsets of the given set. If a given set has population N, its power set has population 2^N.
Proof: Thus obviously the Axiom of the Power set is true for finite sets. The as of yet unproven part is that it is also true for infinite sets. For the infinite case, suppose that S is a set with an infinitely great population. Let W be that to which every individual belongs if & only if he, she, it is either the empty set, the given set S itself, or one of the subsets of S. No member of W can be "simpler" than the empty set, nor more "complicated" than S itself. This mere fact is ipso facto presumptive evidence in favor of the conjecture that W is bounded on all sides on which it is populated.
Also we can assume that W is a compled total reality, lacking nothing essential to its "perfection". Every contribution that an individual makes to the population size is individually discrete and, and all such contributions and equal and intrinsically indivisible provided that neither dies nor reproduces. Therefore the size of W is merely its population. The assumption that W is not a completed total reality entails that W is lacking certain individuals which do happen to meet the necessary and sufficient conditions for membership in W: which is contary to hypothesis!
For all these reasons, W is a set. QED
Corollary: No theory of 'sets' in which the axiom of the power set is denied for infinite sets can accomplish our desired objectives any more than if that axiom had been denied for finite sets.
Thesis 12: Recapitulation of Axiom #6. Axiom of Choice: Unto every given species of differing and mutually exclusive subspecies, even if they are not all at the same level of specificity, there is yet another species containing exactly one member of each such subspecies of the given species, but no other members except members of the such subspecies of the given species.
Thesis 12a: If a given species of different and mutually exclusive subspecies is a set, well then, every possible species containing exactly one member from each subspecies of the given species, but no other members except members of the subspecies of the given species, is also a set. Proof: Suppose that M is a species having different (and mutually exclusive) subspecies. Well then let Q be a species containing one and only one member from each subspecies s of M, but no other members except members of the subspecies of M. Only one member m.s of species Q corresponds to the subspecies s: any formula F for finding m.s from s is a correspondence rule from M to Q according to which two different members of Q cannot correspond to the same subspecies of M: therefore that formula is a function governing the subspecies of M. Also Q does not have more than one member of s, or else any 2 of those two members would be both in Q and both of s - contrary to hypothesis. And every member of M must be one of the members of the species in M. But the subspecies of M are mutually exclusive: meaning that no member of Q can belong to two such different subspecies in common. The same member of Q cannot correspond to two different "members" of M. But now suppose that M is a set. But then by thesis 5, the possible species R of values of m.s is a set, and different m.s implies different s, and the same m.s implies the same s. The possible values of m.s are all contained in Q, therefore every member of R is contained in Q. Also every member of Q is contained in R. Let K be a possible value of m.s. Then it is in species Q. But now let G be in Q. Then by definition it has got to be a member of one of the subspecies of M. Therefore G has to be m.t where t is a subspecies of M; therefore G is in R. Therefore every member of Q is contained in R. Therefore R and Q are exactly identical. Therefore Q itself is a set! QED
Corollary: No theory of sets in which the axiom of choice is denied can accomplish our desired objectives.
Last edited by Casus_Conscientiae on Tue Jun 16, 2026 10:59 pm, edited 2 times in total.
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Casus_Conscientiae
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Re: Fundamental Theses of Set Theory
There Is No Set Big Enough to Contain All Sets; Nor Can Any Set Contain All Other Sets
Thesis 13: Principle of Indifference to Subspeciation. If a given species of different subspecies is a set, well then it will remain a set regardless of how it is classified into subspecies; and if a given species of different subspecies is not a set; well then no change in the manner of subspeciation will ever cause it to become a set. Proof: We will prove the first part.
We have defined some species S by means of some well-defined property φ(x): i.e. x belong to the species S if and only φ(x) = True. We now assume S is a set. According to the definition, this means that we can assume that the species S is a completed actual distinct unbroken unified whole, the size of which is merely the total unbroken whole number of individuals belonging to the species, without resulting in any logical self-contradictions. But then no matter how the species S is subdivided into subspecies, the resulting species S* to which every individual belongs if and only if he, she, or it belongs to at least one of those subspecies is precisely S itself!
But then S remains a set regardless of how S is classified into subspecies. Or else, suppose that one could find a scheme of sub-classification of S into more specifical subspecies such that the resulting subspecies S* to which every individual belongs if and only if he, she, or it belongs to at least one of those subspecies is no longer a set. But then the assumption that S* is a completed actual distinct unbroken unified whole, the size of which is merely the total unbroken whole number of individuals belonging to S* is doomed to result in some logical self-contradiction somewhere. Yet the total population of a species cannot be changed merely by changing how it is subdivided into subspecies. Nor can a mere change of how it is subdivided into subspecies result in the addition of individuals nor the removal of individuals, nor their interchange.
But Theses 1 to 10 imply all of the nine fundamental central axioms of ZFC set theory! The thing that disables the assumption that a given species is a completed actual distinct unbroken unified whole, the size of which is merely the total unbroken whole number of individuals belonging to the species from resulting in any logical self-contradictions does not consist in its subspeciation; for no amount of subspeciation can change whether or not a given individual is a member of S: therefore no amount of change of the manner of subspeciation of S can change the total population of S.
No amount of subspeciation can change the fact that S is a completed actual distinct unbroken unified whole, the size of which is merely the total unbroken whole number of individuals belonging to the species. Subspeciation is a priori dependent upon human choice. We can choose either to subspeciate S either naturally or artificially; and artificial subspeciation is intrinsically contingent upon human choice, even if natural subspeciation is objective and independent of human choice. But the question of whether a species is a set or not is objective and independent of human choice: it is solely a question of the objective truths of logic. Also the question of whether or not to subspeciate S at all into subspecies is also dependent upon human choice.
Thus the assumption that our method M0 of subspeciating S causes S to no longer be a set is impossible: for even without the axioms of the ZFC set theory, it is evident that the question of whether or not to use the method M0 or some other different scheme of subspeciation is inherently dependent upon human freewill! Therefore S will remain a set regardless of how we choose to subspeciate S!
Now for the second part: We now assume that S wasn't originally a set. According to the definition, that means that the assumption that S is a completed actual distinct unbroken unified whole, the size of which is merely the total unbroken whole number of individuals belonging to S is doomed to result in some logical self-contradiction somewhere. Now assume that the mere fact of some particular subspeciation M0 of S into subspecies turns S into a set. But it was remarked earlier that the question of whether or not to subspeciate S at all is dependent upon human choice. We can choose either to accept nature's way of dividing S into subspecies, or use an artificial method: but all such artificial methods are inherently dependent upon human freewill. Therefore the question of whether or not to apply M0 to S is dependent upon human choice: therefore the question of whether or not S is a set would depend upon human freewill: which was already remarked to be impossible, because the question whether a species is a set or not is objective and independent of human choice: it is solely a question of the objective truths of logic.
Therefore even the application of M0 to S cannot cause S to be a set!
Therefore both parts have been proven. QED
Thesis 14: In any two given species, the first of which is a set and the second of which is fully contained by some third species of only all the possible values of some function governing the individual members of the first of the two given species, the second is also a set. Proof: Suppose that we are given the set S, and a species T which is fully contained by some third species U of only all possible values of the function *f* governing the individual members of S. But then according to the axiom of replacement, U = the species to which every individual belongs if & only if he, she, or it is one of the possible corresponding values of *f* governing the individual members of S; therefore U is a set. But T is a subspecies of the set U, therefore T is a set! QED
Thesis 15: In any two given species in both of which, the members of the first of these two species can be put into some one to one correspondence with the members of the second, such that in the one to one correspondence, every individual member of the first species has exactly one and only one corresponding counterpart of the second species and every individual member of the second species corresponds with one and only one member of the first species, the first is a set if and only if the second is also a set. Proof: Suppose that we are given two species S, T such that it is the members of S that can be but into a one to one correspondence with the members of T, such that in this one to one correspondence, every individual member *m* of S has one and only one member *F*.*m* of T corresponding to S and every individual member *m* of T corresponds to one and only member *n* of S. Then *F* is a function governing the individual members of S such that for every member *x* of S, there is exactly one and only one member *y* of T such that *y* = *F*(*x*) and for every member *y* of T, there is exactly and only one member *x* of S such that *y* = *F*(*x*). But then *F* is invertible: thus *x* = *F*⁻¹(*y*). But then let U, = the species to which every individual belongs if & only if he, she, or it is one of the possible corresponding values of *F* governing the individual members of S, and let V = the species to which every individual belongs if & only if he, she, or it is one of the possible corresponding values of *F*⁻¹ governing the individual members of T. Then by construction, every possible value of *F*(*x*) in which *x* ∈ S, belongs ex vi termini to U: therefore S is fully contained in U. It follows by parity of reason that T is fully contained in V!
Now let S be a set. But then according to the axiom of replacement, U is a set. Yet T is a subspecies of U: therefore T is a set. By parity of reason it follows that if T is a set, U must also be a set, and yet S is a subspecies of U, therefore S is itself a set! Therefore S is a set if and only if T is a set. QED
Corollary: In any two species with equal populations, the first is a set if and only if the second is also a set: for if S, T are species with equal populations, we can use the axiom of choice to choose for each individual member of S, exactly one member of T in such a way that every individual member of S has exactly one and only corresponding counterpart member of T, and every individual member of T corresponds with one and only one member of S.
Thesis 16: That to which everyone or everything belongs if and only if he, she, or it is a set cannot itself be a set, nor is there any set big enough to contain all sets. Proof: Or else suppose that the species Sʹ to which everyone or everything belongs if and only if he, she, or it is a set had been a set, which we will call T. Thus x is a member of the species T if and only if x is a set. But then consider the power set P(T) of T. It is evident that P(T) contains T. But P(T) also contains members none of which are identical to T. Also, P(T) contains the empty set. We can now use the axiom of replacement and the axiom of choice to define a function J which has the following properties:
J replaces every fragment g of T with one and only one element J.g which is neither identical to T (indeed, for no individual is a species);
Every two different fragments g, h of T are replaced respectively with only different elements J.g, J.h (neither of which is identical to T).
Thus in the function J, the two individuals J.g, J.h cannot be identical unless g, h are identical.
But P(T) is a set, therefore that, Pʹ(T) which remains of P(T) when T itself is excluded is also a set; but then the effect of J upon Pʹ(T) will be to produce a species Qʹ(T) whose members are: for each and every one of the fragments g none of which are identical to T itself, the individual J.g, which is not identical to T. It follows from Thesis 13 that Qʹ(T) is a set.
But then let Q(T) = the union of P(T) and Qʹ(T). Then by the axiom of union, Q(T) is a set which contains P(T) and also for each and every one of the fragments g none of which are identical to T itself, the individual J.g, which is not identical to T. But because of these extra elements each of which is mutually exclusive with T, Q(T) and T are not identical. Yet because Q(T) is a set, Q(T) must be contained within T; and therefore Q(T) and T would have to be identical, which is impossible, because it was shown that Q(T) contains not only T, but also elements none of which are identical to T at all! Therefore Sʹ is not a set!
Nor is there any set big enough to contain all sets. Or else suppose that such a set could exist: we will call it S_0. But then consider the power set P(S_0), which doesn't just contain S_0, but also contains members none of the which are identical to S_0, and even contains the empty set. We can now use the axiom of replacement and the axiom of choice to define a function J which has the following properties:
J replaces every fragment g of S_0 with one and only one element J.g which is neither identical to S_0 (indeed, for no individual is a species);
Every two different fragments g, h of S_0 are replaced respectively with only different elements J.g, J.h (neither of which is identical to S_0).
Thus in the function J, the two individuals J.g, J.h cannot be identical unless g, h are identical.
But P(S_0) is a set, therefore that, Pʹ(S_0) which remains of P(S_0) when S_0 itself is excluded is also a set; but then the effect of J upon Pʹ(S_0) will be to produce a species Qʹ(S_0) whose members are: for each and every one of the fragments g none of which are identical to S_0 itself, the individual J.g, which is not identical to S_0. It follows from Thesis 13 that Qʹ(S_0) is a set.
But then let Q(S_0) = the union of P(S_0) and Qʹ(S_0). Then by the axiom of union, Q(S_0) is a set which contains P(S_0) and also for each and every one of the fragments g none of which are identical to S_0 itself, the individual J.g, which is not identical to S_0. But because of these extra elements each of which is mutually exclusive with S_0, Q(S_0) and S_0 are not identical. Yet because Q(S_0) is a set, Q(S_0) must be contained within S_0; and therefore Q(S_0) and S_0 would have to be identical, which is impossible, because it was shown that Q(S_0) contains not only S_0, but also elements none of which are identical to S_0 at all! Therefore S_0 is not a set!
Therefore both parts of this theorem have been proven. QED
Corollary I: That to which everyone or everything belongs if and only if he, she, or it is a set has an infinitely great population; no populated species which fails to meet the definition of a set can have only a finitely great population.
Corollary II: There is no set whose individual members are all the other sets; or else suppose that such a set S_0 existed whose individual members are all the other sets. But then U = species to which every individual belongs if and only if he, she, or is it one of the sets other than S_0. Yet it follows by construction that U is a subspecies of S_0 and is therefore a set. But then let {S_0} = that set whose only individual member is S_0. Then let G = the union of {S_0} and U. Then by the axiom of union G is a set. But then G contains as its elements, S_0 and all of the other sets than S_0: therefore G would be a set whose individual members are respectively all possible sets: which according to thesis 16 is impossible!
Corollary III: Nor is there a set which contains all the other sets; or else suppose that such a set S_0 existed that contained all other sets. But then S_0 and all OTHER sets must belong to a common species T, to which every individual belongs if and only if he, she, or it is a set! But T itself is not a set. It is evident that by construction that whatever belongs either to S_0 or to all other sets, ipso facto belongs to T; and whatsoever belongs to T must be a set, and therefore must be either S_0 or one of all the other sets. But S_0 contains all other sets, therefore S_0 contains the species U to which every every individual belongs if and only if he, she, or is a set other than S_0. Yet because S_0 was assumed to be a set, it therefore follows from the axiom of deflation, that U is also a set! S_0 also contains S_0. Every individual belongs to T if and only if he, she, or it belongs either to S_0 or to U. Since S_0 and U are sets, it follows by the axiom of union, that T is also a set (and is the union of S_0 and U): which accoridng to Thesis 16, is impossible!
Thesis 13: Principle of Indifference to Subspeciation. If a given species of different subspecies is a set, well then it will remain a set regardless of how it is classified into subspecies; and if a given species of different subspecies is not a set; well then no change in the manner of subspeciation will ever cause it to become a set. Proof: We will prove the first part.
We have defined some species S by means of some well-defined property φ(x): i.e. x belong to the species S if and only φ(x) = True. We now assume S is a set. According to the definition, this means that we can assume that the species S is a completed actual distinct unbroken unified whole, the size of which is merely the total unbroken whole number of individuals belonging to the species, without resulting in any logical self-contradictions. But then no matter how the species S is subdivided into subspecies, the resulting species S* to which every individual belongs if and only if he, she, or it belongs to at least one of those subspecies is precisely S itself!
But then S remains a set regardless of how S is classified into subspecies. Or else, suppose that one could find a scheme of sub-classification of S into more specifical subspecies such that the resulting subspecies S* to which every individual belongs if and only if he, she, or it belongs to at least one of those subspecies is no longer a set. But then the assumption that S* is a completed actual distinct unbroken unified whole, the size of which is merely the total unbroken whole number of individuals belonging to S* is doomed to result in some logical self-contradiction somewhere. Yet the total population of a species cannot be changed merely by changing how it is subdivided into subspecies. Nor can a mere change of how it is subdivided into subspecies result in the addition of individuals nor the removal of individuals, nor their interchange.
But Theses 1 to 10 imply all of the nine fundamental central axioms of ZFC set theory! The thing that disables the assumption that a given species is a completed actual distinct unbroken unified whole, the size of which is merely the total unbroken whole number of individuals belonging to the species from resulting in any logical self-contradictions does not consist in its subspeciation; for no amount of subspeciation can change whether or not a given individual is a member of S: therefore no amount of change of the manner of subspeciation of S can change the total population of S.
No amount of subspeciation can change the fact that S is a completed actual distinct unbroken unified whole, the size of which is merely the total unbroken whole number of individuals belonging to the species. Subspeciation is a priori dependent upon human choice. We can choose either to subspeciate S either naturally or artificially; and artificial subspeciation is intrinsically contingent upon human choice, even if natural subspeciation is objective and independent of human choice. But the question of whether a species is a set or not is objective and independent of human choice: it is solely a question of the objective truths of logic. Also the question of whether or not to subspeciate S at all into subspecies is also dependent upon human choice.
Thus the assumption that our method M0 of subspeciating S causes S to no longer be a set is impossible: for even without the axioms of the ZFC set theory, it is evident that the question of whether or not to use the method M0 or some other different scheme of subspeciation is inherently dependent upon human freewill! Therefore S will remain a set regardless of how we choose to subspeciate S!
Now for the second part: We now assume that S wasn't originally a set. According to the definition, that means that the assumption that S is a completed actual distinct unbroken unified whole, the size of which is merely the total unbroken whole number of individuals belonging to S is doomed to result in some logical self-contradiction somewhere. Now assume that the mere fact of some particular subspeciation M0 of S into subspecies turns S into a set. But it was remarked earlier that the question of whether or not to subspeciate S at all is dependent upon human choice. We can choose either to accept nature's way of dividing S into subspecies, or use an artificial method: but all such artificial methods are inherently dependent upon human freewill. Therefore the question of whether or not to apply M0 to S is dependent upon human choice: therefore the question of whether or not S is a set would depend upon human freewill: which was already remarked to be impossible, because the question whether a species is a set or not is objective and independent of human choice: it is solely a question of the objective truths of logic.
Therefore even the application of M0 to S cannot cause S to be a set!
Therefore both parts have been proven. QED
Thesis 14: In any two given species, the first of which is a set and the second of which is fully contained by some third species of only all the possible values of some function governing the individual members of the first of the two given species, the second is also a set. Proof: Suppose that we are given the set S, and a species T which is fully contained by some third species U of only all possible values of the function *f* governing the individual members of S. But then according to the axiom of replacement, U = the species to which every individual belongs if & only if he, she, or it is one of the possible corresponding values of *f* governing the individual members of S; therefore U is a set. But T is a subspecies of the set U, therefore T is a set! QED
Thesis 15: In any two given species in both of which, the members of the first of these two species can be put into some one to one correspondence with the members of the second, such that in the one to one correspondence, every individual member of the first species has exactly one and only one corresponding counterpart of the second species and every individual member of the second species corresponds with one and only one member of the first species, the first is a set if and only if the second is also a set. Proof: Suppose that we are given two species S, T such that it is the members of S that can be but into a one to one correspondence with the members of T, such that in this one to one correspondence, every individual member *m* of S has one and only one member *F*.*m* of T corresponding to S and every individual member *m* of T corresponds to one and only member *n* of S. Then *F* is a function governing the individual members of S such that for every member *x* of S, there is exactly one and only one member *y* of T such that *y* = *F*(*x*) and for every member *y* of T, there is exactly and only one member *x* of S such that *y* = *F*(*x*). But then *F* is invertible: thus *x* = *F*⁻¹(*y*). But then let U, = the species to which every individual belongs if & only if he, she, or it is one of the possible corresponding values of *F* governing the individual members of S, and let V = the species to which every individual belongs if & only if he, she, or it is one of the possible corresponding values of *F*⁻¹ governing the individual members of T. Then by construction, every possible value of *F*(*x*) in which *x* ∈ S, belongs ex vi termini to U: therefore S is fully contained in U. It follows by parity of reason that T is fully contained in V!
Now let S be a set. But then according to the axiom of replacement, U is a set. Yet T is a subspecies of U: therefore T is a set. By parity of reason it follows that if T is a set, U must also be a set, and yet S is a subspecies of U, therefore S is itself a set! Therefore S is a set if and only if T is a set. QED
Corollary: In any two species with equal populations, the first is a set if and only if the second is also a set: for if S, T are species with equal populations, we can use the axiom of choice to choose for each individual member of S, exactly one member of T in such a way that every individual member of S has exactly one and only corresponding counterpart member of T, and every individual member of T corresponds with one and only one member of S.
Thesis 16: That to which everyone or everything belongs if and only if he, she, or it is a set cannot itself be a set, nor is there any set big enough to contain all sets. Proof: Or else suppose that the species Sʹ to which everyone or everything belongs if and only if he, she, or it is a set had been a set, which we will call T. Thus x is a member of the species T if and only if x is a set. But then consider the power set P(T) of T. It is evident that P(T) contains T. But P(T) also contains members none of which are identical to T. Also, P(T) contains the empty set. We can now use the axiom of replacement and the axiom of choice to define a function J which has the following properties:
J replaces every fragment g of T with one and only one element J.g which is neither identical to T (indeed, for no individual is a species);
Every two different fragments g, h of T are replaced respectively with only different elements J.g, J.h (neither of which is identical to T).
Thus in the function J, the two individuals J.g, J.h cannot be identical unless g, h are identical.
But P(T) is a set, therefore that, Pʹ(T) which remains of P(T) when T itself is excluded is also a set; but then the effect of J upon Pʹ(T) will be to produce a species Qʹ(T) whose members are: for each and every one of the fragments g none of which are identical to T itself, the individual J.g, which is not identical to T. It follows from Thesis 13 that Qʹ(T) is a set.
But then let Q(T) = the union of P(T) and Qʹ(T). Then by the axiom of union, Q(T) is a set which contains P(T) and also for each and every one of the fragments g none of which are identical to T itself, the individual J.g, which is not identical to T. But because of these extra elements each of which is mutually exclusive with T, Q(T) and T are not identical. Yet because Q(T) is a set, Q(T) must be contained within T; and therefore Q(T) and T would have to be identical, which is impossible, because it was shown that Q(T) contains not only T, but also elements none of which are identical to T at all! Therefore Sʹ is not a set!
Nor is there any set big enough to contain all sets. Or else suppose that such a set could exist: we will call it S_0. But then consider the power set P(S_0), which doesn't just contain S_0, but also contains members none of the which are identical to S_0, and even contains the empty set. We can now use the axiom of replacement and the axiom of choice to define a function J which has the following properties:
J replaces every fragment g of S_0 with one and only one element J.g which is neither identical to S_0 (indeed, for no individual is a species);
Every two different fragments g, h of S_0 are replaced respectively with only different elements J.g, J.h (neither of which is identical to S_0).
Thus in the function J, the two individuals J.g, J.h cannot be identical unless g, h are identical.
But P(S_0) is a set, therefore that, Pʹ(S_0) which remains of P(S_0) when S_0 itself is excluded is also a set; but then the effect of J upon Pʹ(S_0) will be to produce a species Qʹ(S_0) whose members are: for each and every one of the fragments g none of which are identical to S_0 itself, the individual J.g, which is not identical to S_0. It follows from Thesis 13 that Qʹ(S_0) is a set.
But then let Q(S_0) = the union of P(S_0) and Qʹ(S_0). Then by the axiom of union, Q(S_0) is a set which contains P(S_0) and also for each and every one of the fragments g none of which are identical to S_0 itself, the individual J.g, which is not identical to S_0. But because of these extra elements each of which is mutually exclusive with S_0, Q(S_0) and S_0 are not identical. Yet because Q(S_0) is a set, Q(S_0) must be contained within S_0; and therefore Q(S_0) and S_0 would have to be identical, which is impossible, because it was shown that Q(S_0) contains not only S_0, but also elements none of which are identical to S_0 at all! Therefore S_0 is not a set!
Therefore both parts of this theorem have been proven. QED
Corollary I: That to which everyone or everything belongs if and only if he, she, or it is a set has an infinitely great population; no populated species which fails to meet the definition of a set can have only a finitely great population.
Corollary II: There is no set whose individual members are all the other sets; or else suppose that such a set S_0 existed whose individual members are all the other sets. But then U = species to which every individual belongs if and only if he, she, or is it one of the sets other than S_0. Yet it follows by construction that U is a subspecies of S_0 and is therefore a set. But then let {S_0} = that set whose only individual member is S_0. Then let G = the union of {S_0} and U. Then by the axiom of union G is a set. But then G contains as its elements, S_0 and all of the other sets than S_0: therefore G would be a set whose individual members are respectively all possible sets: which according to thesis 16 is impossible!
Corollary III: Nor is there a set which contains all the other sets; or else suppose that such a set S_0 existed that contained all other sets. But then S_0 and all OTHER sets must belong to a common species T, to which every individual belongs if and only if he, she, or it is a set! But T itself is not a set. It is evident that by construction that whatever belongs either to S_0 or to all other sets, ipso facto belongs to T; and whatsoever belongs to T must be a set, and therefore must be either S_0 or one of all the other sets. But S_0 contains all other sets, therefore S_0 contains the species U to which every every individual belongs if and only if he, she, or is a set other than S_0. Yet because S_0 was assumed to be a set, it therefore follows from the axiom of deflation, that U is also a set! S_0 also contains S_0. Every individual belongs to T if and only if he, she, or it belongs either to S_0 or to U. Since S_0 and U are sets, it follows by the axiom of union, that T is also a set (and is the union of S_0 and U): which accoridng to Thesis 16, is impossible!
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nyquistfrequency
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Re: Fundamental Theses of Set Theory
You are doing too much work, you could have just simply said in a single sentence:
A set of all sets refutes itself through regress as there could be no "first set" which contains all other sets, as that first set, being a set by definition must be contained by "a set of all sets".
A set of all sets refutes itself through regress as there could be no "first set" which contains all other sets, as that first set, being a set by definition must be contained by "a set of all sets".