CANTOR DEBUNKED

[Anonymous No.15606277]

Let there be a mapping between the natural numbers and finite strings of math symbols, many of which form a real number. We can from here show an injection from the reals onto the naturals, or simply delete all the sequences of math symbols that don’t form numbers and show a bijection between the naturals and reals.

And now you say: “But we can use the decimal forms of these numbers and construct a diagonal number that’s not in the list.” And at first glance, that seems true. However, notice that this definition of the diagonal number is not a simple sequence of math symbols denoting sums, exponents, limits, etc. And so its finite definition differs in structure from the finite definitions from our original list. But we can include such types of finite definitions in the list as well. So that now, no matter what number you define in finite terms, it will necessarily be included in the list. Therefore the definition says “construct a number that is different from all numbers in this list,” but the list includes itself, so the number is different than itself, a contradiction. Therefore such a definition is invalid.

So we see that the strongest form of the diagonal argument fundamentally rests on a paradox (contradiction) of self-reference.

*Note that we ignore numbers that aren’t definable in finite terms, as these numbers are not necessary to satisfy the real number axioms, and they do not exist anyway since we have no way of interacting with them*

[Anonymous No.15606304]

https://en.m.wikipedia.org/wiki/Richard%27s_paradox

Sweaty, this has been deboonked by fact checkers.

[Anonymous No.15606333]

>>15606304
> The preceding paragraph is an expression in English that unambiguously defines a real number r. Thus r must be one of the numbers rn. However, r was constructed so that it cannot equal any of the rn (thus, r is an undefinable number). This is the paradoxical contradiction.
It’s a contradiction because the definition references itself and defines a number that is different from itself. That’s the paradox. That’s the contradiction. The definition is invalid.

[Anonymous No.15606365]

To avoid the English language and the concern of “meta-mathematics,” we only need a system that allows us to talk about mappings, as well as distinguish strings that don’t form numbers from those that do. Then there is no ambiguity in how numbers are constructed. The ONLY problem is self-reference, as this is illogical and contradictory. So we remove it from the system. Then it follows that there are as many reals as natural numbers, and there is no such thing as uncountable sets.

[Anonymous No.15606450]

> Skolem's paradox is that every countable axiomatisation of set theory in first-order logic, if it is consistent, has a model that is countable. This appears contradictory because it is possible to prove, from those same axioms, a sentence that intuitively says (or that precisely says in the standard model of the theory) that there exist sets that are not countable. Thus the seeming contradiction is that a model that is itself countable, and which therefore contains only countable sets, satisfies the first-order sentence that intuitively states "there are uncountable sets".
Now we see that there is no paradox at all, because the diagonal argument itself is the paradox. There is no such thing as uncountability

[Anonymous No.15606465]

>>15606450
The problem in Skolem's paradox is first order logic which also gives rise to a shitload of other unnecessary paradoxes. Conveniently in second order logic many of these problems don't exist anymore.

[Anonymous No.15606557]

>>15606450
>There is no such thing as uncountability
What real number comes after 0?

[Anonymous No.15606567]

>>15606557
The rational numbers are countable but you could still ask the same question about them. Please refrain from posting any more in this thread.

[Anonymous No.15606573]

>>15606557
Bro just construct a well-ordering. Like bro just use axiom of choice.

[Anonymous No.15606637]

>b-but Cantor’s argument was different!
Cantor’s argument relies on the same self-reference since he starts by assuming that all real numbers are in the list. So the diagonal number’s existence is a contradiction: it is both in the list and not in the list. Rightly, Cantor suspected that there was something wrong with assumptions. But instead of doubting his faulty construction of the diagonal number using self-reference, he doubted the list itself. But if you simply exclude illogical and contradictory self-reference from mathematics, we would not have such problems.

[Anonymous No.15606643]

Further proof that the diagonal argument is wrong:
> Cantor’s paradox is based on an application of Cantor’s theorem. Cantor’s theorem states that given any finite or infinite set
S
S
, the power set of
S
S
has strictly larger cardinality (greater size) than
S
S
. The theorem is proved by a form of diagonalisation, the same idea underlying Richard’s paradox. Cantor’s paradox considers the set of all sets. Let us call this set the universal set and denote it by
U
U
. The power set of
U
U
is denoted
P
(
U
)
P
(
U
)
. Since
U
U
contains all sets it will in particular contain all elements of
P
(
U
)
P
(
U
)
. Thus
P
(
U
)
P
(
U
)
must be a subset of
U
U
and must thus have a cardinality (size) which is less than or equal to the cardinality of
U
U
. However, this immediately contradicts Cantor’s theorem.

[Anonymous No.15606686]

>>15606277
Just in case you're being serious.

You can construct a mapping between the natural numebrs and finite strings of maths simbols, only under the assumtion that there is a countable ammount of symbols. I have no problem assuming that. However, I have an issue with the following sentence.

> We can from here show an injection from the reals onto the naturals, or simply delete all the sequences of math symbols that don’t form numbers and show a bijection between the naturals and reals.

Your logic is sound but you're assuming that every real number can be written as a finite string of maths symbols. That is a very strong assumption which would show, indeed, that real numbers are countable. Can you provide a proof that every real number can be written as a finite string of maths symbols (where those symbols belong to a countable family)? Also, can you provide a definition of real numbers too? Just so I know we're referring to the same mathematical object.

The second paragraph also relies on the same assumption.

[Anonymous No.15606706]

>>15606686
The real numbers follow certain axioms as well as the least upper bound property. As for the finitude of their definitions, see the note at the bottom of the OP.

[Anonymous No.15606806]

>>15606706
In your note you said:

> Note that we ignore numbers that aren’t definable in finite terms, as these numbers are not necessary to satisfy the real number axioms, and they do not exist anyway since we have no way of interacting with them.

If you call D the set of real numbers that are definable in finite terms, D is a countable set as you proved correctly. You may call D the set of definable real numbers.

You may argue that we should restrict ourselves to the set of definable real numbers because we will never interact with them. Your belief is a valid one, which I do not share.

I asked for a definition of real numbers because _it does not follow_ from the usual axioms of the real numbers that every real number is definable, since **definable numbers do not have the least upper bound property**. If that bothers you, don't use the real numbers. But do not claim that real numbers are countable.

[Anonymous No.15606830]

>>15606806
If you want a proof of the last claim, 'definable numbers do not have the least upper bound property', it's not very hard.

First, real numbers are uncountable. It is easy to prove that any ordered field with the least upper bound property is isomorphic to the real numbers. In particular, there would be a bijection between the field and the real numbers, proving that such field is uncountable.

The set D is a field: 0 and 1 are both definable. If a and b are two numbers in D, 'a + b' can be written with a finite amount of symbols, so we can add elements in D. Similar work shows that every field axioms hold for D, so D is a field.

D is ordered: trivial, since D is a subset of the real numbers, which are ordered.

If D had the least upper bound property, it would follow that D is uncountable. Since D is countable, D does not have the least upper bound property.

[Anonymous No.15606831]

>>15606806
If a number is undefinable, then you have no way of proving that it is a necessary element of the real numbers to satisfy the axioms and least upper bound property. You could try to argue that the real numbers are closed under addition, and that undefinable numbers can be produced by infinite additions made on other real numbers, but the closure property is defined in finite terms.
> *definable numbers do not have the least upper bound property**.
Definable real numbers have the least upper bound property, yes.


But even if we accepted undefinable numbers as real numbers, I’m not sure that this would prevent the contradictory self-referential construction from occurring. As I said >>15606637, Cantor’s argument still implicitly uses self-reference. That is why it leads to a paradox here >>15606643. Self-reference is almost always the cause of a paradox. It should be abolished from mathematics, as well as undefinable numbers, as we have no use for them.

[Anonymous No.15606839]

>>15606830
> First, real numbers are uncountable
If we agreed on this, then you wouldn’t even have to continue with this proof, retard.

[Anonymous No.15607425]

>>15606839
It is not a matter of agreement. I can show that the real numbers are uncountable in Lean or Coq. You didn't prove that all real numbers are definable. You didn't prove that definable numbers have the least upper bound property.

By the way, I think you should get down of your high horse if you are being serious and really believe that you're right. Cantor's proof is over one hundred years old. It has been proven over and over by maths professors in universities. There are many different proofs of uncontability of the real numbers (like https://math.stackexchange.com/a/491942 which doesn't mention decimals at all). It has been proven with formal theorem provers like Lean (https://leanprover-community.github.io/mathlib_docs/data/real/cardinality.html#cardinal.not_countable_real)) and Coq (https://stackoverflow.com/q/51933107).). And yet you believe that, with a three paragraph argument you are more knowledgeable than everyone else. Alright.

[Anonymous No.15607444]

>>15607425
The point is that everyone is working within the wrong math foundations. Predicate mathematics is superior