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Results tagged with soft-question
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user 1946
Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. Do not use this tag for easy or supposedly easy mathematical questions.
4
votes
Heuristic interpretations of the PA-unprovability of Goodstein's Theorem
To my way of thinking, the arguments you mention seem not to distinguish sufficiently between the content of Goodstein's theorem as a universal claim $\forall p$ and Goodstein's theorem as a collectio …
- 236k
14
votes
Why is inner model theory evidence for consistency of large cardinals?
The explanation is philosophical rather than mathematical.
The idea is simply that the inner-model theory provides a rich account of what it would be like for the large cardinal axioms to be true, and …
- 236k
1
vote
Papers on history and philosophy of mathematics suitable for master's students
For the philosophy of mathematics, I wrote my book specifically with mathematical readers in mind. Many readers have told me that they appreciate the accessible manner the book has of treating even su …
- 236k
10
votes
Accepted
Why include $0$ and $1$ in the signature of Presburger arithmetic?
It is the same in Peano arithmetic, where the standard language is $\{+,\cdot,0,1,<\}$ for the standard model $\langle\mathbb{N},+,\cdot,0,1,<\rangle$, even though $0$, $1$, and $<$ are definable from …
- 236k
31
votes
What could be some potentially useful mathematical databases?
There are several natural examples from set theory.
Here is a database on consequences of and equivalent formulations of the axiom of choice, which is searchable by keyword and axiom form, and which …
- 21.6k
105
votes
Accepted
Have you solved problems in your sleep?
On several occasions it has happened that I have made a key insight while sleeping or drifting in and out of sleep.
For example, one of the critical ideas in my paper
Joel David Hamkins, Gap forcing, …
- 4,713
9
votes
Accepted
Is there a well defined subset of the integers that cannot be defined as a property of a rec...
Over at my answer to I. J. Kennedy's question about degrees of irrationality, I described several hierarchies of definable complexity that transcend computability. I have copied my answer below. Alrea …
- 1,112
39
votes
True by accident (and therefore not amenable to proof)
Apart from your specific example, the idea of
truth-by-accident has been studied in the context of formal
first-order languages, which includes the language of graph
theory, and in his dissertation, K …
- 18.4k
104
votes
Theorems with unexpected conclusions
My favorite example of this phenomenon is Goodstein's Theorem.
Take any positive number $a_2$, such as the number $73$, and write it in complete base $2$, which means write it as a sum of powers of $2 …
- 4,713
1
vote
Are there interesting examples of theorems proved using ‘height’ extensions?
Here is another example.
The maximality principle in forcing is the scheme asserting of every statement $\varphi$ in the language of set theory that if there is forcing extension $V[G]$ of the set-the …
- 236k
6
votes
Accepted
Are there interesting examples of theorems proved using ‘height’ extensions?
Here is another instance, which appears in my recent paper with Bokai Yao on second-order reflection in the context of KMU with abundant urelements.
Joel David Hamkins and Bokai Yao, Reflection in se …
- 236k
5
votes
Decision problems for which it is unknown whether they are decidable
It remains open whether the won-position problem of infinite chess is decidable, the problem of determining whether a given finite position in infinite chess is winning for white or not. See Richard S …
- 236k
32
votes
What are some examples of colorful language in serious mathematics papers?
In this MO answer, I mentioned Arnold Miller's lecture
notes, where he gives
an entertaining account of the MM proof system (for Micky Mouse), having as axioms all validities and modus ponens as the o …
- 4,713
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 …
- 82.7k
21
votes
Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be...
I believe that there are many instances of this phenomenon in set theory, where an elaborate theory is developed over a period of years by many people, even though the theory is not viewed ultimately …
- 236k