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Results tagged with lo.logic
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user 3106
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.
9
votes
What does it mean for a mathematical statement to be true?
Part of the reason for the confusion here is that the word "true" is sometimes used informally, and at other times it is used as a technical mathematical term.
Informally, asserting that "X is true" …
2
votes
Proof of Gödel incompleteness
One place in the proof where Jech uses the finiteness of $S$ is the point where he passes from "$M \vDash \sigma$ if and only if $N \vDash (m \vDash \sigma)$" to $N \vDash (m$ is a model$)$". To make …
- 82.7k
8
votes
Formulas for the liar paradox
As Joel David Hamkins said, the standard answer to your question is that formal languages like the first-order language of arithmetic cannot express the liar paradox because they cannot express the pr …
- 82.7k
16
votes
Why do people say Gödel's sentence is true when it is true in some models but false in others?
There are several distinct issues to be aware of.
First, there is the question of the meaning of "true in a model" versus being simply "true" (or "true simpliciter" if you like Latin). Once one reali …
- 82.7k
68
votes
Situation with Artemov's paper?
The essential issue is the same as one that has been discussed many times here on MO, for example here and here. Consider the following string $S$.
$$(\exists x \exists y \exists z : xxx + yyy - zzz = …
- 82.7k
15
votes
What is the reverse mathematical strength of the fundamental theorem of algebra?
Bjørn Kjos-Hanssen has answered the stated question but I think it would help to make a few clarifying comments.
Let BWQ denote the statement, "Every bounded infinite sequence from $\mathbb Q$ has an …
- 82.7k
5
votes
When do we get $CON(ZF)$ in transfinite progressions of consistency statements?
There is no known explicit combinatorial description of the proof-theoretic ordinal of ZFC. Even much weaker set theories have so far defied explicit description. For a recent account that gives some …
- 82.7k
5
votes
Replacing logician-constructive with combinatorist-constructive?
I think the closest thing to what you are looking for are the logical systems studied in Cook and Nguyen's recent book Logical Foundations of Proof Complexity. These are systems in which the provably …
- 82.7k
5
votes
Independence of PA implies independence of PA union all true $\Pi_1$ statements
As others have pointed out, the assertion is false. What the authors mean by calling it "folklore" is that virtually all the known techniques for proving a "natural" statement (like the Paris-Harring …
- 82.7k
13
votes
What are the differences between Woodin and Sy Friedman regarding set theoretic truths?
It is only a half-joke to say that for a layman, there is no difference between their approaches. Take the following question for example: Does there exist an uncountable subset of $\mathbb R$ that c …
- 82.7k
15
votes
Circular, or missing, definition in set theory?
I'm not sure I understand your question, since at first it sounds like you're thinking of $\in$ as a multi-valued function that sends a set $A$ to an element $x$ of $A$, but then I would expect you to …
- 82.7k
7
votes
Logical equivalences for FTA
Pete Clark has an expository paper, Factorization in integral domains, that is relevant to your question. In the particular case of FTA, I think it's more fruitful to ask for conditions on a commutat …
- 82.7k
7
votes
"Introduction to mathematical logic" book from a formalist perspective
I'd like to elaborate a bit on the suggestion, mentioned by some others, that your complaint is not really about platonism versus formalism, but about the common practice of being not completely clear …
11
votes
What does T+non-Cons(T) mean?
While I don't disagree with the substance of what Qiaochu Yuan and Andrés Caicedo have said, I'm not happy with the terms "gibberish" or "useless."
It's important to bear in mind that when we say "con …
- 82.7k
17
votes
What can be proven in Peano arithmetic but not Heyting arithmetic?
According to Harvey Friedman, the following theorem is provable in PA but not HA:
Every polynomial $P:\mathbb{Z}^n \to \mathbb{Z}^m$ with integer coefficients assumes a value closest to the origin …
- 82.7k