Cantor's Diagonalization Refutes Set Theory!

[nyquistfrequency]

Cantor's diagonalization although meaningful and rigorous is greatly misinterpreted.

Since an endless decimal sequence can by definition have no end, means it cannot be completed by definition.

Yet since it doesn't stop at any given point, it must be an endless indefinite becoming, an inexhaustible process of succession, same with any such endless list, an endless list is not a completed thing, but an inexhaustible process.

But there must be something simultaneous which conditions the process, that which governs the process' becoming, the purpose or telos of the process if you will. For otherwise there would be nothing which would condition the next term in the sequence.

So what is actually happening here is that we are trying to represent something (for instance a finite magnitude like the diagonal of a square), which cannot be represented as a numeral, since these are things of different incommensurable orders of being. 

The secret of incommensurability and orders of being is that succession cannot reach that simultaneity which governs the succession, the "how" cannot reach the "why". For instance, the succession of additions $x+x+x+...$  can never reach the principle of multiplication which governs all those additions simultaneously so one can never reach $x^2$ by adding $x$ to itself. It is only true where $2\cdot2 = 2+2$ so the in terms of qualitative Pythagorean numbers, the dyad the very distinction between substance (how) and essence (why) is the only level where succession and simultaneity are unified.

So an overflow happens, we cannot properly express this continuous (simultaneous) magnitude of the square root of two in our discontinuous (successive) base 10 system so in endlessly exceeding the capacity to be expressed in this way it turns into an endless approach without completion, an indefinite succession of approximations, a becoming rather than a being.

And this is the reason we have change, because the Absolute which is non-limited cannot be identical to itself in any finite way, for otherwise it would be a smallest part which is impossible, so any finite perception creates an overflow of being into becoming, an overflow of eternity into time.

In this way, Cantor's diagonalization only shows the impossibility of a process of all processes, that all processes is not any particular process. Just as Damascius said "All things" cannot include the Principle as a "thing", so "All processes" cannot be a "process" on the list.

Like diagonalization is a direct self negation, a self denial, the clearest case of reductio ad absurdum if I've ever seen one:

1.  We assume that such a "number" should be in this list by the way we have defined it as the "list of all Reals between 1 and 0"
2.  Use diagonalization to construct a "real number" that is deliberately altered at every n-th decimal place to ensure it differs from every "number" on the list.
3. Therefore it is proven to not be in the list, which says it should include that "number".
4.  Therefore the conclusion should be "such a list is impossible in that it has denied its own possibility" NOT "there are 'greater infinities'" or whatever that means.

It's not that the "number" or rather process is not on the list it's that the concept of such a list is impossible to begin with. Yet for some reason the mathematicians, these so-called champions of logic, can't seem to grasp a reductio ad absurdum?

Cantor's diagonalization is a perfect example of both why one cannot reach continuity through discontinuity as in zeno's paradoxes (as a list of all possible reals which would bridge continuity and discontinuity is an impossibility) and that the indefinite or endless is always incomplete and therefore less than the finite, as it is not the principle but only its expression.

It's a very useful proof to point to when those who say that Zeno's paradoxes has supposedly been solved by calculus. But still integration does prove that an endless approach can be complete, it only shows that there is something being approached, that there is a "why" guiding the "how".
This means that all endless becomings or processes can be expressed as something finite. The endless is less than the finite for the endless will always be incomplete.

This completely compromises ZFC Set Theory's "Axiom of Infinity", exposes the illusion of the cardinals, and restores the mereological fallacy where the whole cannot be derived from the part.

[StudentDriver]

"But there must be something simultaneous which conditions the process, that which governs the process' becoming, the purpose or telos of the process if you will. For otherwise there would be nothing which would condition the next term in the sequence."

- that's me. I condition the next term in the sequence. LMao

[nyquistfrequency]

"But there must be something simultaneous which conditions the process, that which governs the process' becoming, the purpose or telos of the process if you will. For otherwise there would be nothing which would condition the next term in the sequence."

- that's me. I condition the next term in the sequence. LMao

But what is "I"? Who are you?
If you can not describe exactly what you are, you cannot validly claim that you condition anything. For one's nature may be such to make any conditioning impossible.

[StudentDriver]

Crazy ahhhh question bro. You should know what "I" means. Many people had asked that question, and this question is still relevant today. Look around. You getting a little spiritual broski? You shouldn't get into woo woo Bulls*t

You keep asking questions like that, the malevolent forces gonna get yo ahhh

[evening-calm]

By any chance have you come across the mathematician Norman Wildberger and his reformulation of mathematics on the rational numbers deliberately excluding the reals critical of their ‘logical’ constructions?

There’s a whole syllabus developed there but there’s a few debates in which he’s participated which summarize his world view echoing I suspect some of the critique above.

[nyquistfrequency]

By any chance have you come across the mathematician Norman Wildberger and his reformulation of mathematics on the rational numbers deliberately excluding the reals critical of their ‘logical’ constructions?

There’s a whole syllabus developed there but there’s a few debates in which he’s participated which summarize his world view echoing I suspect some of the critique above.

I do know of Norman Wildberger, however he is a constructionist, and my view of math aligns with Rene Guenon, Platonists, and Ancient Greek Geometers more than it does with his view. However the constructionists are right in their view of the formalists, and they are right that anything constructed must be real (otherwise they wouldn't be able to be constructed), however there are things which cannot be constructed which are nonetheless real which are proven via negativa. I have not seen Wildberger's interpretation of diagonalization, this is entirely my own critique.

If you want to know more about this read Guenon's The Metaphysical Principles of the Infinitesimal Calculus.