TonesInDeepFreeze

That is not specification of an order, let alone of "THE inherent order". Of course, we could move left to right and also up and down, zigzagging to s...

I'm not sure, but I think a place to look might be proofs showing the relative consistency of anti-wellfoundeness. Maybe there is a construction in su...

Without the axiom of regularity, you can't prove ~Ex x = {x} = {{x}} = {{{x}}} ... for as finitely many iterations you want to make. As I said, if you...

Yes, there is an analogy and similarity, but also the modification used makes a great difference too.

x = {x} Read my previous post regarding proof and lack of proof of existence.

I too will plead, as at this time, especially as I am rusty in the subject, I wouldn't be able to marshal enough knowledge to explicate the details of...

I could only do that by using anti-foundation (or non-wellfounded) set theory. These are different: (1) The theory proves that there is not a set that...

See sections 4.4 and 4.5 here: https://plato.stanford.edu/entries/goedel-incompleteness/ For philosophical concerns about mathemtics see: https://www....

No, there are two kinds of proof involving contradiction: assume ~P derive contraction conclude P assume P derive contradiction conclude ~P The first ...

That doesn't sound right to me. The properties that are checked are syntactical. What semantical properties does Wikipedia claim are checked?

The incompleteness proof is intuitionistically valid and does not require excluded middle.

You said that I have difficulty stating the theorem. Stating a proof of the theorem is more than stating the theorem. I stated the theorem without dif...

There is no rule of set theory that we must name a set with set abstraction notation. Moreover, for a finite set with members that are members of them...

Yes, analogues.

I don't take exception to the Goldstein quote. But her book about incompleteness needs to be read critically. As I recall (though I can't cite specifi...

No. (1) The Godel sentence is not an equation. (2) "rules of math" is unclear. (3) We don't look outside the "rules of math" even given a reasonable u...

Yes, your quote from the book is well taken. And it is clear in the context of the book, but some people might not realize that context, so I just wan...

Aside from whatever SophistiCat might say, it is not the case that formalism regards the incompleteness theorem in that way. (1) Sentences are not tru...

To be clear, Franzen is taking exception to the theorem being incorrectly co-opted in many of those context.

"self-reference" used pejoratively in reference to Godel's theorem is a red herring. The self-reference is seen by looking outside the object language...

What you think about it is one matter. (What you think about it is based on a collection of confusions and misunderstandings you have.) But you said t...

Why don't you look up a text in set theory so you would know how set theory axiomatically, clearly and unambiguously proves theorems and defines terms...

"member of itself twice" has no apparent mathematical meaning.

But a couple of points: It's not inconsistent. It does not imply a contradiction. A contradiction is a statement and its negation. If you claim to poi...

Wrong. It is possible that all these are the case: x = {x y z} (so x is a member of itself), and y is a member of y, and y is a member of x, and z a m...

Set theory doesn't prove things in this kind of context by saying "the task cannot be completed". However, the axiom of regularity disallows infinite ...

You didn't go far enough in the argument: If there is a set of all sets, then it has the subset that is the set of all sets that are not members of th...

Usually, we have an intuitive notion that sets are not members of themselves. However, since 'is a member' is primitive, we will not have a formal exp...

With regularity, there is a set whose members are all and only those sets that are members of themselves. That set is the empty set. And if we drop re...

No, you know it. (1) is set theory proving there is no set whose members are all and only those sets that are not members of themselves. (2) is Tarski...

I don't like using markup. The text is plenty clear enough.

I didn't. No, I'm not violating separation. Separation and extensionality were used to prove the existence of 0 (in an axiomatization where the existe...

Are you serious? Come on, you know how to do it yourself: ZF |- ~ExAy(yex <-> ~yey) ZF |/- ~ExAy(yex <-> yey). ZF-R |/- ~ExAy(yex <-> yey) ZF |- ExAy(...

Russell's paradox shows the contradiction in set theory with unrestricted comprehension. After Russell's note, we moved to a set theory that does not ...

Looks okay now. No, I don't. Indeed, I said the opposite in my original post. Howzabout you quote me where you think I claimed that there exists a set...

Yes, when 'we' includes you. But with math, we do specify specific kinds of order. The notion of infinite sets is used calculus, which is mathematics ...

'before' and 'after' are often in a temporal sense, but clearly not exclusively. Not in English. And surely not in math that doesn't mention temporali...

Your math notation in your previous post does not format form me. Anyway, {x | ~xex} is not at question. There is no such set.

I don't think you mean {x | ~xex}. We're talking about {x | xex}. And yes, without regularity, we can't prove there is a nonempty set {x | xex}. But t...

That article gives an answer. It's a great article. (By the way, Panu Raattkainen is a top notch source on the subject.)

With regularity, It's the empty set. And we can't derive a contradiction by dropping an axiom, so such a set is consistent also without regularity. Bu...

Maybe I need to be double-checked, but my reasoning tells me that undecidability follows right from incompleteness.

It is terrible. I mentioned why earlier in this thread.

You mentioned the benefit of a course in logic. In another thread, I have listed what I consider to be the best textbooks leading to the incompletenes...

That is incorrect. Of course, to countenance sets being member of themselves, we have to delete the axiom of regularity. With that done: Suppose xex, ...

If you can correctly extract from the video that Godel's argument is circular, then the video is wrong. We should learn Godel's argument from a carefu...

I told you twice what Godel's theorem is: Instead of recognizing that, you bring up a different matter. You will not make any progress here if you can...

You have so many misconceptions. But let's take one thing at a time, starting here: Do you recognize that I did state Godel's theorem?

correction: When I said 'total linear ordering' I should have said 'strict linear ordering'.

I read fine. But with your lack of replies to many crucial points, I admit that I can't read what doesn't exist. Here is a previous post tracking your...