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Mar
15
answered What advantage humans have over computers in mathematics?
Mar
3
answered Interpretation of the Second Incompleteness Theorem
Mar
2
comment Relationship between first and second incompleteness theorems
@Dave, all things end, I guess.
Mar
2
comment Relationship between first and second incompleteness theorems
@Robin. I do think you should be able to generalize my part, but am not sure how it meshes with Emil's construction.
Mar
1
revised Relationship between first and second incompleteness theorems
added 65 characters in body
Mar
1
comment Relationship between first and second incompleteness theorems
Sorry, bad editing. I meant the list was the usual list, but fpa was not assuming PA3. I'll edit to make this clear.
Feb
29
answered Relationship between first and second incompleteness theorems
Jul
16
answered Which model of computation is “the best”?
Jan
18
awarded  Yearling
Jan
11
awarded  Nice Question
Sep
30
comment Why have mathematicians used differential equations to model nature instead of difference equations
"(That highest number plus one is zero.)" This has always struck me as a curious view. What is gained by supposing the highest number plus one is zero, rather than supposing the highest number plus one does not exist? Indeed, what is gained by supposing there is a highest number, rather than just being agnostic about the matter (and just not assuming the axiom "every number has a successor")?
Mar
15
comment Is there a theory of abuse of notation?
Is the poster looking for sufficient or necessary conditions?
Jan
18
awarded  Yearling
Nov
2
comment Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic
@EN. Many thanks for your comments.
Nov
1
comment Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic
Also, if I may, one other line of question (I think this interests many many more people than myself, so it is why I am presuming to take up your time). Is the notion of "actual number" different from the notion of "natural number"? If the answer is yes, does induction hold for the actual numbers? If the answer is no, why the use of "actual" instead of "natural"?
Nov
1
comment Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic
@EN. Why not? For example it would take only around 13 tries to bifurcate 5000 successfully. First question: do you see no reason to believe that ψ(80^i) is a theorem of F if i = 2500? If you answer yes, second question has i = 1250; and if you answer no, second question has i = 3750. etc. Is it a question of vagueness, like 5000 hairs is a lot of hair, but you would say there is no least i such that i hairs is a lot of hairs because "lot" is vague?
Nov
1
comment Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic
Is there a least i such that you see no reason to believe that ψ(80^i) is a theorem of F?
Oct
21
comment Decidability of equality of elementary expressions
Is it possible to give an answer to the easier question where i is replaced by 1, i.e. one considers as expressions: 1; exp(x); ln(x); (x⋅y) ?
Sep
30
comment Has anyone pursued Frege's idea of numbers as second-order concepts?
Suppose there are ten objects, so you can prove the second-order numbers up to 10 exist, are different, and occur in a successoring chain. Let P be the third-order object containing the primes less than 10. Then the claim that there are four of them is the claim there exists a second order set A and a third-order relationship R such that R is 1-1 onto from P to A, where #A = 4. (Perhaps I'm getting mixed up in the orders here, but I hope that works.)
Sep
24
comment Has anyone pursued Frege's idea of numbers as second-order concepts?
In arithmetic without the successor axiom, you could say it in this fashion. If S(S(S(S(S(S(S(S(S(S(0))...) exists, then the number of primes less than it is 4.