TonesInDeepFreeze

The axiom of infinity is how we prove that there is a set that has every natural number as a member. From the axiom of infinity, we derive that there ...

A typo there? I think you meant 'z in x iff z in y'?

There's a point from a while back. Maybe we can fix it. I said that ExAy y e x is consistent. You disputed that. So I pointed out that I am not saying...

Exactly. That goes right with what I've been saying. Without sarcasm I say that it gives me a good feeling that reason, intellectual curiosity and com...

I know you're kidding. But underneath there lies an actual point for me, which is that I don't think you know how insulting you are in certain threads...

Did you mean for that to be in the 'Infinity' thread? In that thread, you've now seen that I already had given you the Enderton pages yesterday and I ...

Nope. I am consistent with the SEP article. The context in this discussion is plain predicate logic where substitution works, not intensional contexts...

That might be. I'm speaking in broad terms about them in that regard. If the article draws a needed distinction then I should say that they are at lea...

If you mean that it would help for my posts to link to yours, then I'll hope not to forget doing that each time. My preference regarding you is that y...

That is where he gives the semantics for '=', as I mentioned that '=' is given a fixed interpretation.

Yes, I said that it doesn't mention set theory, but rather it is a place to see the logical axiom schema for first order logic with equality. However,...

And you can look at the SEP article 'Identity' where you'll see: Leibniz’s Law, the principle of the indiscernibility of identicals, that if x is iden...

I hope you know that 'the crank' does not refer to you. If that was not clear in the context, then I should have made it clear.

I already gave you the pages: And I gave you page number and line numbers for Shoenfield: I said that the indiscernibility of identicals is formalized...

Seeing just that one phrase from the great song made my night. Such a soul satisfyingly beautiful song by a gigantically great composer.

I'm talking about interpretations for languages as discussed in mathematical logic. There are uncountably many sets, so there are uncountably many uni...

I gave you a very detailed answer. I can't do better than what I already wrote. Or, if you like, let me know what you don't understand in my post.

Exactly.

I don't want to watch a video right now.

We're talking about different things. I'm talking about formal theories and interpretations of their languages as discussed in mathematical logic, and...

Of course, w is a limit ordinal, and it is the ordinal limit of the sequence of all the natural numbers. But, just to be clear, we still need to prove...

That exchange deserves nothing more than a snort. The crank still can't vindicate his claims sets by answering what is the inherent order of the set w...

No wiggling. It was faulty of me to reference that book without specifying that I do not claim it discusses the identity axioms. I had previously admi...

The premises don't not specify that the button is ever pushed. The premises do not specify that there are only two states, unless, in this very hypoth...

Making clear corrections, giving generous explanations, commenting the deplorable methods of cranks, and posting ideas in general is not ranting. / Lo...

Mostly, I would be very interested to see your proof of: (x = y & y = z) -> x = z You may use only the law of identity Ax x = x and the axioms of set ...

I addressed that in detail. You could reread what I wrote. I post for at least as an end in and of itself, and also meaningful record for whomever may...

I don't see it as a confusion of Michael. He is only rendering Thomson's setup. And I don't see Michael getting tripped up by the metaphorical use of ...

EDIT LATER: Disregard this post. I hope to post a revision. I'll try this: Suppose: There are two states F and N. At any moment either F is active or ...

It doesn't matter to me what the lamp is. I can regard the problem abstractly, in terms just of: time two states sufficient and necessary conditions f...

(1) Why not use 11:00 rather than 10:00? Usually the problem concerns 11:00 to 12:00, which is tidy for the halvings of the durations. (I'll use 11:00...

I don't recall the context in which I recommended Enerton's set theory book, but if it was about first order logic with identity for set theory, then ...

Enderton's set theory text is a great book. But, as with many excellent set theory books, it doesn't mention all the technical details. I didn't say t...

If that is considered a form of reductio ad absurdum, then every proof of a negation is proof by a form of reductio ad absurdum. In a natural deductio...

It's garden variety modus tollens: If there is a bijection then there is a surjection There is no surjection. Therefore, there is no bijection. No nee...

Thank you.

C1 is a premise. It is the premise that the lamp has only two states. But that's not a substantive problem; only that I'm mentioning that it is a prem...

You are not including the premise "The lamp can only be on if immediately preceding it was off. And the lamp can be off only if immediately preceding ...

It rejects that having an on/off state is determined by an immediate predecessor state, so when we reject that premise, it is not ruled out that the s...

You can couch the hypothetical situation with whatever premises you like. In that sense it's not a matter of me agreeing or disagreeing. And the part ...

We reject that it is possible for (1) (2) (3) to hold together. So we can reject (1) and be left with a consistent set of two premises. So it is not r...

Classical mathematics itself first formulated that there is no algorithm that prints all the members of an infinite set and halts. How nice. My point ...

Constructivism is broader than intuitionism. Intuitionism is one form of constructivism. I don't opine as to what other poster's notion of constructiv...

But, if I am not mistaken, your argument comes down to: From the assumption that (1) (2) (3) are together possible, we infer that time is not infinite...

And you see now that a reductio argument is not needed; indeed Cantor did not use a reductio argument.

g is a list of denumerable binary sequences, and we construct a denumerable binary sequence not listed by g. Or if reals are addressed: g is a list of...

That is incorrect. In the instance about "can", I merely provided you the information that mathematics doesn't need to use "can" but rather can use "i...

You hadn't said that you understand the point, so the point deserved repeating. But you don't have to keep repeating that point, as it has many times ...

That seems to drawing an inference from an impossibility. If we agree that (1) (2) (3) are together impossible, then we can infer anything from the as...

A horse can push a cart, not only pull it. I haven't refused that. But I suggest that 'immediate predecessor' is a good way of couching the matter. Th...