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Results tagged with lo.logic
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user 1946
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.
393
votes
What are some reasonable-sounding statements that are independent of ZFC?
"If a set X is smaller in cardinality than another set Y, then X has fewer subsets than Y."
Althought the statement sounds obvious, it is actually independent of ZFC. The statement follows from the …
241
votes
Accepted
Is the analysis as taught in universities in fact the analysis of definable numbers?
The concept of definable real number, although seemingly
easy to reason with at first, is actually laden with subtle
metamathematical dangers to which both your question and
the Wikipedia article to w …
- 236k
174
votes
Accepted
Solutions to the Continuum Hypothesis
Since you have already linked to some of the contemporary
primary sources, where of course the full accounts of those
views can be found, let me interpret your question as a
request for summary accoun …
- 236k
173
votes
Most 'unintuitive' application of the Axiom of Choice?
I have enjoyed the other answers very much. But perhaps it
would be desirable to balance the discussion somewhat with
a counterpoint, by mentioning a few of the
counter-intuitive situations that can o …
148
votes
Accepted
Nontrivial theorems with trivial proofs
Bertrand Russell proved that the general set-formation principle known as the Comprehension Principle, which asserts that for any property $\varphi$ one may form the set $\lbrace\ x \mid \varphi(x)\ \ …
144
votes
Accepted
Reductio ad absurdum or the contrapositive?
Although the other answers correctly explain the basic logical equivalence of the two proof methods, I believe an important point has been missed:
With good reason, we mathematicians prefer a direct …
- 236k
116
votes
Why worry about the axiom of choice?
Yes, many people continue to fuss about the Axiom of Choice.
At least part of the explanation for why people continue to fuss as they do over the Axiom of Choice is surely the historical fact that the …
79
votes
What are some reasonable-sounding statements that are independent of ZFC?
"There is no definable well-ordering of the real numbers."
Although many mathematicians simply believe this statement to be true, actually, it is independent of ZFC. In Goedel's constructible univer …
74
votes
Accepted
What's wrong with the surreals?
At a recent conference in Paris on Philosophy and Model Theory (at which I also spoke), Philip Ehrlich gave a fascinating talk on the surreal numbers and new developments, showcasing it as unifying ma …
- 236k
72
votes
Nontrivial theorems with trivial proofs
Cantor proved that the set of real numbers is uncountable---it cannot be put in bijective correspondence with the natural numbers---but the proof is a simple diagonalization: if the real numbers could …
70
votes
Accepted
A remark of Connes on non-standard analysis
...as soon as you have a non-standard number, you get a non-measurable set.
Every nonstandard natural number $N$ gives rise to a nonprincipal ultrafilter $U$ on $\mathbb{N}$, by saying that a set $X …
- 236k
67
votes
Knuth's intuition that Goldbach might be unprovable
You are right to view the Goldbach conjecture as having a particularly simple logical form. Such statements of the form "for every $n$, property $P(n)$ holds", where $P$ is a particularly simple state …
- 236k
66
votes
Clearing misconceptions: Defining "is a model of ZFC" in ZFC
Here is a result along the lines you are requesting, which
I find beautifully paradoxical.
Theorem. Every model of ZFC has an element that is a
model of ZFC. That is, every $M\models ZFC$ has an elem …
- 236k
62
votes
Accepted
Zorn's lemma: old friend or historical relic?
I agree with almost everything in your post. But still, I believe I know why people use Zorn's lemma.
My answer. Zorn's lemma encapsulates succinctly many of the consequences of AC via transfinite rec …
- 236k
52
votes
Checkmate in $\omega$ moves?
Update. (Oct 28, 2015) See below, for a position with game value $\omega^4$.
This is a great question, which I have been pondering for some
time.
I have just completed a joint article Transfinit …
- 236k