# Tagged Questions

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### On Counted Languages

In my recent question on Godel Completeness I mentioned that there was a related question I wanted to ask, but would keep separate. I have been recently studying "non-well ordered sets" and Chapter 7 ...
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### Constructiveness of Proof of Godel's Completeness Theorem

As a mathematician interested in novel applications I am trying to gain a deeper understanding of (the non-constructiveness of) Godel's Completeness Theorem and have recently studying two texts: ...
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### Bourbaki and AC: How does he proves ZL?

In the book Set theory, Chapter 3 N.Bourbaki, I would like to understand how Bourbaki proves ZL. I wrote the proof. It uses Zermelo's principle (which is okay since they are equivalent), so I tried to ...
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### Plausibility argument for Zorn's Lemma

In "Mathematical Physics" by Robert Geroch, the following 'plausibility argument' is given for Zorn's Lemma [If every totally ordered subset of a partially ordered set $S$ is bounded above, $S$ has a ...
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### Axiom of Choice - Type Theory (Proof)

Background In Intuitionistic Type Theory (p. 27-28), Martin Löf provides a proof of the axiom of choice that is constructively valid. This version is considerably weaker than the ordinary set theory ...
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### Extending a Filter in a Well-Ordered Boolean Algebra to an Ultrafilter WITHOUT the Axiom of Choice

Hypothesis: Let $B$ be a well-ordered boolean algebra and let $F \subseteq B$ be a filter on $B$. Goal: Show that $F$ can be extended to an ultrafilter without the axiom of choice (or any equivalent ...
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### Well-ordering theorem and second-order logic

I am confused by this sentence in the Wikipedia article for "Well-ordering theorem": ...the well-ordering theorem is equivalent to the axiom of choice, in the sense that either one together with ...
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### Boolean prime ideal theorem and the axiom of choice

The Boolean prime ideal theorem is strictly stronger than ZF, and strictly weaker than ZFC. I'm looking for nice examples (like the existence of non-measurable set) that request at least that theorem ...
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### Logic: Teichmüller-Tukey Lemma and the Axiom of Choice

How can you proof that the Teichmüller-Tukey Lemma (which says that if $S$ is nonempty and of finite character, $S$ contains a maximal element with respect to the subset ordering), implies the Axiom ...
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### Dependent choice does not imply “the reals are well-ordered”; citation?

As silly as this sounds, I can't find a proof that the axiom of dependent choice (DC) does not imply that the reals are well-orderable. My memory is that this is a fairly early result in the history ...
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### Axiom of dependent choice and $\aleph_1$

Assume we have to make a construction on countable sets, which requires choice. If we need to repeat the same construction up to cardinal $\aleph_1$ (for example to construct a chain of elementary ...
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### Is there a deductive system for second-order logic that is complete with respect to Henkin semantics?

I have heard that second-order logic with Henkin semantics is a lot like first-order logic. Does this mean it has a complete deductive system? If so, what's an example of such a deductive system, and ...
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### Extending our language with a new function symbol

Given an arbitray first-order theory (not necessarily a set theory) and definable predicates $P(*)$ and $Q(*,*)$ in the language of that theory, if we adjoin a new function symbol $f$ together with ...
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### If $B\not\prec A$, does $A\preceq {\cal P}(B)$ or $A\preceq {\cal P}{\cal P}(B)$, etc in ${\sf ZF}$?

I've heard it said that theorems based on choice are also available in ZF "a few powersets away", and I think this is one of them, but I'm not sure how to prove it. (I'm also interested to hear of ...
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### Truth Value of Theorems in Axiomatic Set Theory

I encountered set theory these past couple of days in discrete mathematics, and my professor was talking about the axiom of choice and ZFC. He said that depending on which axiom you started from, you ...
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### Infinite Set is Disjoint Union of Two Infinite Sets

A finite set is a set such that there exists a bijection from it to some finite ordinal. An infinite set is a set that is not finite. In ZF, can you prove that every infinite set is the union of two ...