TonesInDeepFreeze

I previously corrected that: The set of real numbers between 1 and 2 has the same cardinality as the set of real numbers between 1 and 3. The set of r...

The above post is correct and mentions a crucial point. Therefore, it will be of no interest to too many people.

So your method of conversation is to ignore when someone informs you nearly a dozen times on a point: In this context, mathematics doesn't use 'infini...

Nothing at all like it is.

Again, in mathematics, the concept is 'is infinite' as an adjective, not 'infinity' as a noun. And continuity is a different idea, while the idea of "...

In mathematics, with or without the axiom of extensionality, '=' means identity.

Of course notions of infinity pertain in different areas of study. But just to bear in mind, the original post is a challenge to the idea that there a...

So your qualification is noted. But what you wrote originally naturally would be taken as a pejorative. "just mental showboating" would not ordinarily...

A person hasn't studied the pertinent mathematics, doesn't know anything about it, doesn't understand it. So their response to it is to say that it mi...

To add to the above: If X := X->Y then X <-> (X->Y). But we don' t have the converse that if X <-> (X->Y) then X := X->Y. So X := X->Y is not equivale...

Notation I'll use: 'iff' for 'if and only if' df. x is equinumerous with y iff there is a one-to-one correspondence between x and y df. x is countable...

To put your musings in perspective, here are the mathematical facts: The set of rational numbers between any two natural numbers is not sequenced by t...

What evidence do you have that fishfry left because of this thread? The reason technical content should not be shunted elsewhere is that if the philos...

I said I don't have anything immediate to say about subjective impressions of mathematics. I am not thereby like a "chatbot" that is not interested in...

It's not complicated. Definitions: f is an injection iff f is a one-to-one function. We may also say 'f is an injective function'. f is an injection f...

Posts are missing from this thread, including some of my own. What happened? And a post of mine was deleted in another thread. So I listed the reasons...

Note that 'countable' in mathematics does not mean that any human being can count every member of the set. Rather, 'countable' in mathematics only mea...

For about the sixth time, and this is one of the points you keep refusing to address: I don't begrudge anyone from having whatever concept and definit...

I said exactly what points you are not addressing, and now you just come back to insist that you are addressing them though you are not. And you said,...

I can say it is an informal notation for the set of all and only the natural numbers because that is exactly what it is an informal notation for. The ...

I'm no pot, since I haven't ignored what the poster said. But you are such a pot, as recently you ignored what I said.

Mark Nyquist: Now for the third time: Anyone can have whatever concept of mathematics they want to have. But having a different concept from set theor...

Philospher19, again ignored for the second time what I wrote about "contradiction".

Another misconception: "saying 1,2,3,4 ad infinitum or {1,2,3,4,...} does not mean one has shown an infinite number of natural numbers." Yes, saying t...

I am paying close attention to what you are saying. While I just now showed specifically and fundamentally how you skipped what I said. And if you wan...

Then you believe an untruth. And for about the fifth time, there is no object named by 'infinity'. So there is no object named by 'the quantity of inf...

Time well spent would be to learn some mathematics rather than claiming untrue things about it. Anyway, did someone say "beyond infinity"? 'Beyond inf...

If one claimed that the definition of 'equinumerous' must lead to a definition of cardinal subtraction, moreover a requirement that a definition of ca...

Someone said 'This sentence is false' doesn't indicate what sentence is being referred to. In 'This sentence is false', 'this sentence' is referring t...

What invalid statement was implied by mathematics formalized in first order logic? By definition, all non-logical axioms are invalid (i.e. not validit...

When people argue that the paradox is explained away by saying that the sentence is not meaningful, they are overlooking at least this: In formalized ...

It helps because talking about "minus" in that way is incorrect and leads to confusions. If we are talking with people unfamiliar with the arithmetic ...

The framework is arbitrary just as definitions in general are arbitrary. One can have whatever formalization of mathematics one wants to have and any ...

We don't use 'less' in that sense with infinite sets. Rather, if x is an infinite set, and y is a set with n number of elements, then there are n numb...

Again, it is not meaningful to say 'infinity' as if there is an object named by it. Rather, there are various sets that have the property of being inf...

There is no object named 'infinity'. Rather, there is the property of being infinite. There is not a "minus" operation involving infinite cardinals in...

There is no definition of cardinal subtraction with infinite cardinals in a way such as with the integers.

Usually, in mathematics we do not use 'infinity' as a noun. There is not an object that we call 'infinity'.* Rather, we use the adjective 'is infinite...

NOTE: Earlier in this thread, with a different username, I used the word 'onto' not intending its mathematical meaning of a function onto a set, but r...

It was claimed that R = {x | x not-e x} has not been explicated. '{ x | }' is the abstraction operator. It is variable binding notation that takes any...

Again, what Russell's paradox shows is that by pure logic: There is no set y whose members are all and only the sets that are not members of themselve...

An argument was made that it is contradictory for a set to have as members all and only the sets that are not members of itself (correct) but that it ...

To properly discuss 'lists' in the context of set theory, we need to have a formalization of the notion of lists: At least personally I would take 'li...

Just to be clear, neither Regularity nor Pairing are needed to prove: There is no y such that for all x, x is a member of y iff x is not a member of x...

Regarding a notion that ZF is "inadequate/incomplete". If ZF is consistent then ZF is incomplete, in the sense that there are sentences in the languag...

There are too many confusions in this thread. This post can be used for correct and definite formulations. (For more, see https://plato.stanford.edu/e...

It is recommended that anyone truly interested in the subject may read a book in mathematical logic (for example, Enderton's 'A Mathematical Introduct...

I wrote here an outline of a proof of Tarski's theorem. But it incorrectly glossed over crucial details. So I have deleted it here.

For the fourth time: The incompleteness theorem pertains only to recursively axiomatizable theories. A set of statements that includes a proper subset...

Tarski says no such thing as claimed two posts above. And, for the third time: The incompleteness theorem pertains only to recursively axiomatizable t...