Questions tagged [arithmetic]
11 questions
16
votes
1 answer
When it is correct to use Tarski's undefinability theorem versus Gödel's incompleteness theorem?
Smullyan (1991, 2001) has argued forcefully that Tarski's undefinability theorem deserves much of the attention garnered by Gödel's incompleteness theorems. That the latter theorems have much to say about all of mathematics and more…
Xodarap
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12
votes
5 answers
How should we characterize the relationship between mathematics and philosophy of mathematics?
How should we characterize the relationship between mathematics and philosophy of mathematics?
Specifically, in what ways might the study of philosophy of mathematics make a mathematician better at his work, and which contributions from philosophy…
wajed
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9
votes
9 answers
Why does division take a lot more mental effort than multiplication?
It is mentally more difficult to divide 2 numbers than it is to multiply them. If you ask me what is 3 * 27, i will immediately tell you 'a little less than 90'.
However, I really have no idea what is 3 / 27. It is much harder to do this operation,…
Dennis Kozevnikoff
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7
votes
3 answers
Can mathematical sentences in different theories be identified?
My question motivated by a part of this page from Saul Kripke's book Naming and Necessity, which is also viewable on google books. In the middle of the page he say something, which seems unnatural to me: Namely he implies that one mathematical…
Nikolaj-K
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5
votes
2 answers
Is there any possible world in which 2+2=5?
Gödel's incompleteness theorems show that arithmetic is either inconsistent or incomplete, and that arithmetic cannot prove its own consistency. It is useful to believe that arithmetic is consistent, and therefore also incomplete, but there are…
Dan Brumleve
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3
votes
1 answer
Good texts on logicism
I'm trying to learn about the logicist programme by myself so I was wondering, what are some good book/papers/articles on logicism? I'm looking for introductory to medium level texts, nothing very deep.
user926356
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3
votes
3 answers
Can we add to PA a new predicate T such that for every sentence A of the old vocabulary the new theory proves T(Godel numeric number of A) iff A
I am new to logic but I believe this is not a difficult problem, yet I am still soo confused, and the reason for that is because there are so many gaps in my knowledge or maybe I have overlooked so many "obvious" argument. I truly appreciate any…
user71346
- 131
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0
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0 answers
Does "mixing up the notation" bring out a synthetic character for basic arithmetic?
There's this thing in the work of Immanuel Kant and Hannah Arendt where they'll slip into Greek and/or Latin, sometimes in the middle of a sentence (even if for just a word there), or sometimes like one sentence in a paragraph will be in one…
Kristian Berry
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0
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1 answer
Are the truths of arithmetic logically necessary?
Are true statements of arithmetic logically necessary? That is, is "2+3=5", the commutativity of addition of natural numbers, and the infinitude of primes, among other statements, logically necessary? For example, I doubt very much that there is a…
user107952
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-1
votes
1 answer
Robinson Arithmetic and Church-Turing Thesis
What is the connection (if any) between proving the undecidability of Robinson Arithmetic and the Church-Turing Thesis? If there is any connection to CTT, is it necessary?
user49323
-6
votes
3 answers
Can we logically derive a value for 0÷0?
I have a "proof" that 0÷0 = 2:
0÷0 = (100 - 100) ÷ (100 - 100)
= (10⋅10 - 10⋅10) ÷ (10⋅10 - 10⋅10)
= (10² - 10²) ÷ 10(10 - 10)
= (10 + 10)(10 - 10) ÷ 10(10 - 10)
= (10 + 10) ÷ 10
= 20 ÷ 10
= 2
It's obviously wrong mathematically (0÷0 is…
Wenura
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