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I'm a graduate student at UC Berkeley, studying mathematical logic. I'm especially interested in reverse mathematics and abstract model theory.
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9h

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Understanding Mathematics
You might also add some more details, and then ask the question at math.stackexchange (which is a different site). 
1d

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Induction and nonstandard halting times of standard machines
No, they should be the same  that's a much clearer way to phrase it. 
1d

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Fun math puzzle
One such place is math.stackexchange  this site is specifically for researchlevel questions (see the FAQs). 
1d

revised 
Induction and nonstandard halting times of standard machines
edited title 
1d

asked  Induction and nonstandard halting times of standard machines 
2d

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Will this be a case of self plagiarism or will it annoy the referee?
This might be a better fit in academia.stackexchange? 
May 18 
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${\frak b}$ and ${\frak d}$ defined with $\leq$ instead of $\leq^*$
You don't need the "$+\aleph_0$," since $\mathfrak{d}$ is infinite. 
May 18 
answered  ${\frak b}$ and ${\frak d}$ defined with $\leq$ instead of $\leq^*$ 
May 15 
accepted  Minimal degrees of structures 
May 15 
revised 
Minimal degrees of structures
added 28 characters in body 
May 15 
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Minimal degrees of structures
For those, like me, who have not heard of pbgenericity before: ac.elscdn.com/S0168007298000141/… 
May 15 
asked  Minimal degrees of structures 
May 15 
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standard deviation of an equation
This question is not appropriate for this forum, which is for research mathematics; math.stackexchange would be a better fit. 
May 15 
revised 
standard deviation of an equation
edited tags 
May 8 
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Knuth's intuition that Goldbach might be unprovable
Well, Andres Caicedo and Terry Tao both give answers which explain why a $\Pi^0_1$ sentence $P$, if undecidable (we need $P$ to be nondisprovable, not unprovable) in ZFC, is true. Moreover, contrary to your answer, no consistency or soundness assumption is needed here  if ZFC is inconsistent then the hypothesis "$P$ is unprovable in ZFC" is never satisfied, and regardless of whether ZFC is sound it cannot fail to prove a true $\Sigma^0_1$ sentence. So the answers I mentioned are completely accurate, and there's no need for further elaboration. 
May 8 
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Knuth's intuition that Goldbach might be unprovable
But this itself is nonabsolute: a still larger model $O\supseteq N$ might contain a bijection between $X$ and the set $N$ thinks is $\mathbb{R}$, and moreover might contain no new reals  so $O$ would think $X$ is not a counterexample to CH. And this can keep going forever. 
May 8 
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Knuth's intuition that Goldbach might be unprovable
Absoluteness is a kind of nonverifiability: let's say I have a model $M$ (nice and wellfounded) of ZFC and a set $X\in M$. Now, $M$ might think $X$ has size continuum, in the sense that $M$ contains a bijection between $X$ and a set $A\in M$ which $M$ thinks is the set of all reals. But, there might be real numbers not in $M$  for instance, if $M$ is countable. So there might be a "larger" model of ZFC, $N\supseteq M$, such that there is no bijection  in $N$  between $X$ and the set $N$ thinks is $\mathbb{R}$. In $N$, $X$ might be a counterexample to CH. 
May 8 
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Knuth's intuition that Goldbach might be unprovable
The point is that a counterexample to Goldbach, if it exists, will be both easy to find and easy to prove a counterexample. By contrast, for something like the Continuum Hypothesis, a counterexample might be (1) impossible to effectively describe, so ZFC wouldn't be able to even refer to it, and (2) impossible to prove a counterexample even if it were definable. Here, we learn nothing from the statement "ZFC doesn't prove CH," since ZFC just doesn't have the ability to build and verify a counterexample if one exists. (This is vague, but hopefully helpful?) 
May 8 
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Knuth's intuition that Goldbach might be unprovable
Well, the point is that (1) a counterexample to Goldbach can be verified: if $a\in\mathbb{N}$ is a counterexample to Goldbach, then $PA$ (in fact, just the ordered semiring axioms) can prove that (the numeral corresponding to) $a$ is a counterexample to Goldbach; and (2) Goldbach quantifies over a set which is appropriately enumerable (the natural numbers), as opposed to a more wild set (such as the set of sets of reals, as the Continuum Hypothesis does). 
May 8 
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Knuth's intuition that Goldbach might be unprovable
That's not quite true  you need the property $P$ to be appropriately simple (barring additional assumptions, this means $\Sigma^0_1$). That is, you need "being a counterexample" to be absolute. 