No, it is right. Yes, set theory is consistent if and only if there is a model of set theory. But if set theory is consistent then set theory itself d...
You mean there is a bijection between the naturals and the computable reals. And I didn't claim otherwise. I only pointed out the incoherence of a par...
I don't suggest that you suggested that the set of noncomputable reals is countable. Meanwhile, I changed my post above to pertain to definability rat...
I changed this post greatly: So to make the argument work that there are only countably many definable real numbers, maybe something like this in a se...
I didn't make that assertion. I only commented on the poster's particular argument. (I later edited 'definable' to 'defined', but in either case, my c...
Right from the start of your argument, the merely ostensive (and not specified by actual mathematical description) list you gave is either actually no...
Using 'infinity' as a noun in the context of cardinality is incorrect and supposed refutations of set theoretic notions of cardinality by using 'infin...
If any of a number of articles (e.g. https://medium.com/cantors-paradise/uncomputable-numbers-ee528830d295) on the Internet are correct, then it is no...
Clearly there is a problem with the manner in which you are reading. My very first sentence in this thread: "One doesn't have to provide much argument...
Would you please slow down. You're swinging your arms around wildly. I didn't refute my own point. I never claimed that a demonstration of the consist...
All that said, please slow down and read exactly what I post at exact face value. That will avoid time wasting squabbles about who said what about wha...
There is nothing cryptic in what I wrote and it is not false without further explanation. And I did not say that there is a universal set. I said that...
Of course it contradicts the axiom of regularity. But I said ONTO ITSELF. The formula ExAy yex is not a contradiction. It is consistent. Trivially, it...
One doesn't have to provide much argument that the following claim onto itself is not self-contradictory: (1) There exists a set such that every set i...
Consider the actual reality (not just a hypothetical possibility) that the mathematicians thoroughly studied the subject matter down to its finest det...
It seems to me that birthdays celebrate the fact that the person was born. The person is closer to death every day, not just on their birthday. I see ...
"FOL is syntactically complete." That is incorrect. It is not the case that for every formula, either FOL proves the formula or FOL proves the negatio...
The completeness theorem, as applied to Q in particular, is that for first order logic, if a formula F is entailed by a set of formulas Gamma, then th...
Russell's paradox applies not only to the element relation, and not just to set theory, but to any 2-place relation whatsoever and to logic in general...
The explanations are both correct and clear. I said it's basic combinatorics. I have no opinion on what you know about it otherwise. I have offered yo...
It is exactly correct, precisely clear, and states exactly how it works in terms of von Neumann ordinals or even using any sets. And x^0 = 1 x^(y+1) =...
To add to that comment, we note that it works with any sets, not just von Neumann ordinals. Take any set S that has x number of elements and any set T...
{f | f is a function & domain(f) = 2 & range(f) is a subset of 2} = { {<0 0> <1 0>} {<0 0> <1 1>} {<0 1> <1 0>} {<0 1> <1 1>} } which is a set having ...
x^y may be defined as the number of functions from y into x. The empty function is the only function from 0 into x, so the number of functions from 0 ...
TF proves there is no such set. But meanwhile set theory proves there is that set. The set is the universe for a model of TF. The set itself is not a ...
(ZF-C)+~C is just ZF+~C. That is not ZF. I’ve never seen that theory discussed (though maybe it comes up somewhere.) I take ‘-‘ to mean ‘without’ and ...
No the part that has the headline: First-order theory of arithmetic That is first order PA. (In this context, By ‘PA’ we mean first order PA.) And PA ...
Here is what I responded to: When you say "its sets", I take it you mean the universes's sets, i.e. the sets that are in the universe. Of course, for ...
I'll use M, because it stands out better. So: For a theory T with 'e' in the language, a formula F(x) in the language for T "invokes" a proper class r...
I mentioned (putting it in these terms now) that for any theory T and any cardinality C, if there is a model M of T, then there is a model M* of T suc...
By the way, since historically different writers formulate class theory differently, for sake of definiteness, I choose one in particular: ‘Model Theo...
We should not overlook that ‘class’ does not mean just ‘proper class’. Some classes are sets and other classes are proper classes. Everything in NBG i...
notation: for a function f, let f’y= the x such that fx =y. Let M be a model for the language L. Let S be any set whatsoever with the same cardinality...
You more or less despise "my world", not knowing anything about it other than it includes people who may recommended books without first knowing wheth...
Because I recommended some printed books (which you may check for yourself whether they are also online or not), you infer that I don't know enough ab...
Whatever may be the demerits of printed books, at least I an tell you that those are exceptionally great books and it is not certain that there will b...
Exactly. I don't know about online, but here is a course of books I highly recommend, in order of study: * Logic: Techniques of Formal Reasoning - Kal...
Yes, as I said, if a theory has an infinite model, then it has both countable and uncountable models. That's a non sequitur. The set of sentences is c...
She's adding a constant to the LANGUAGE of the theory. The model is a model for the language. The model is not a language. There is a language. Then t...
Writers might differ on the definition of 'theory', but it turns roughly the same with whatever adjustments we need:. (1) A theory is a set of sentenc...
What I wrote there (and with similar remarks on this particular point) may be too strict. It belies that there may be a less narrow notion of 'model o...
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