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Results tagged with lo.logic
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user 6794
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
178
votes
Set theories without "junk" theorems?
I apologize for posting as an answer what should really be a comment, connected to one of Jacques Carette's comments on my earlier answer. Unfortunately, this is way too long for a comment. Jacques …
- 73.2k
61
votes
Accepted
What is Realistic Mathematics?
When Solovay showed that ZF + DC + "all sets of reals are Lebesgue measurable" is consistent (assuming ZFC + "there is an inaccessible cardinal" is consistent), there was an expectation among set-theo …
- 73.2k
49
votes
Set theories without "junk" theorems?
Structural set theory, as described on the nlab page you linked to, is probably the best answer to your question. To avoid junk theorems, one must deviate somewhat from ordinary ZF-style set theory w …
- 73.2k
48
votes
Why can't proofs have infinitely many steps?
Even if logic were extended to allow infinitely long proofs, your attempted proof of the countable axiom of choice would still have a gap or two. After the infinitely many steps asserting that there …
- 73.2k
46
votes
Accepted
What can be proven in Peano arithmetic but not Heyting arithmetic?
The first example that occurs to me is (a formalization in the language of arithmetic, via coding, of) "For every Turing machine M and every input x, the computation of M on input x either terminates …
- 73.2k
40
votes
Arguments against large cardinals
Most of the answers have addressed the "consistency" part of the original question, "Why is it so unreasonable to think that the existence of large cardinals contradicts ZFC?" There's another part, a …
- 73.2k
38
votes
Why should we believe in the axiom of regularity?
I think of the axiom of regularity along with the axiom of extensionality as formalizing what I mean by "set". Once upon a time, before paradoxes, one could think of sets as just any collection of th …
- 73.2k
35
votes
Accepted
Do the algebraic integers form a free abelian group?
Pontryagin's criterion says that, for a countable, torsion-free, abelian group to be free, it suffices that every finitely many elements lie in a finitely generated pure subgroup. The rings $\mathcal …
- 73.2k
34
votes
Accepted
Interpretation of the Second Incompleteness Theorem
For the philosophical point encapsulated in (*) in the question, it seems that corollaries of the second incompleteness theorem are more relevant than the theorem itself. If we had doubts about the c …
- 73.2k
33
votes
Accepted
Is P=NP relevant to finding proofs of everyday mathematical propositions?
Let me address the issue at the beginning of the original question: If P=NP were proved and an algorithm with reasonable constants found, would mathematicians stop trying to prove things? The relevan …
- 73.2k
32
votes
Accepted
Is there a name for a family of finite sequences that block all infinite sequences?
Intuitionists use the name "bar" for what you called a blocking set. The relevant context is "bar induction," the principle saying that, if (1) a property has been proved for all elements of a bar an …
- 73.2k
31
votes
About the axiom of choice, the fundamental theorem of algebra, and real numbers
The fundamental theorem of algebra is, unless I miscounted quantifiers, a $\Pi^1_2$ sentence of second-order arithmetic and therefore absolute between the full universe and Gödel's constructible unive …
- 73.2k
30
votes
Finite axiom of choice: how do you prove it from just ZF?
Although the answers already given are correct, let me add some information (essentially just rephrasing the bracketed part of Thomas Scanlon's answer) that I've found useful for students who raised t …
- 73.2k
29
votes
Is PA consistent? do we know it?
As Mirco Mannucci's answer suggests, the alternatives labeled 1) and 2) in the question are (for some people) not mutually exclusive. The consistency of PA indeed "has a proof as valid as any other t …
28
votes
Accepted
Unprovable statements S where the only way to prove S is to assume S
The following is essentially Joel's answer and also essentially the last part of Francois's answer, but its "look and feel" seems different enough to make it worth pointing out. The main point is tha …
- 73.2k