All Questions
Tagged with philosophy logic
383
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Gödel Incompleteness theorems - gap between first order logic and arithmetic
I have a doubt concerning Gödel's incompleteness theorem which might be stupid but that makes me uneasy. If I'm not mistaken, Gödels proof is broadly undertaken using first order formulas, numbering ...
4
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(When) are recursive "definitions" definitions?
This is a "soft" question, but I'm greatly interested in canvassing opinions on it. I don't know whether there is anything like a consensus on the answer. Under what conditions (if any) are ...
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Confused about abstract models for axiomatic systems
I am studying axiomatic systems and I have a hard time understanding how one is supposed to come up with an "abstract" model for an axiomatic system.
I will use the following example taken ...
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statements that can be accepted by finitists.
"On the finitist view, the formula $\exists n P(n)$ is meaningful only when it is used as a statement specifying how to calculate an $n$ for which $P(n)$ is true".
It is mentioned as above ...
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Why do we need a metatheory if we can include self-referencing language in the object theory?
I am wondering why we need to have a metatheory in order to talk about a theory- why can't we just add self-referencing terms to the language of the formal system on which the theory itself is based, ...
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What is the formal system when we are using many different sets of axioms?
I am just starting to learn about formal systems, and have learnt that the many axiom systems in Mathematics, such as those of plane geometry, Peano's axioms, vector axioms, etc. can each be used to ...
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Why is addition not completely defined here?
Say for the natural numbers, we define addition this way:
$0 + 0 = 0$, and if $n+m = x$, then $S(n) + m = n+S(m) = S(x) $
Say we have the regular Peano axioms, except we delete the axiom of ...
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Trying to understand how numbers themselves (s0, ss0, sss0, etc) are represented in Gödel numbering
Problem solved: I did not actually read the table given on page 70 of nagel and newman. s does have a Godel number. It's 7. So ss0 would be broken down into 7, 7, and 6, since 0 is given the number 6. ...
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Are there any problems about the difference between set theoretic definitions of polynomials?
I am a novice about this question, so if there is a misunderstanding then I apologize for it.
As for Peano axioms, if I choose Zermelo natural numbers, and you choose von Neumann ones, then this doesn'...
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1
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doesn't the independency phenomenon make a case for non-classical logic? [closed]
alright, this question is philosophical and somewhat fuzzy. i also admit to knowing little about logic. all in all, this question can possibly be easily resolved by either pointing to (perhaps even ...
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What is the logical system of Tractatus Logico-Philosophicus?
Tractatus Logico-Philosophicus states simply that
6 The general form of the truth function is: $[\bar p, \bar\xi, N(\bar \xi)]$. This is the general form of the sentence.
Wikipedia and other sources ...
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Finitists reject the Axiom of Infinity - are there groups who reject the others?
I've seen rejections of the Axiom of Infinity. This is called finitism. Some ultrafinitists even add the negation of the Axiom of Infinity. Definitely doable.
I've seen rejections of the Axiom of ...
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What did Richard Dedekind mean exactly by his statement about generality?
But—and in this mathematics is distinguished from other sciences—these
extensions of definitions no longer allow scope for arbitrariness; on
the contrary, they follow with compelling necessity from ...
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When mathematicians say "true" do they mean "true in all models"?
According to the comments to this question,
Truth is ordinarily defined by reference to models.
If so, even axioms and theorems are not true without reference to a model.
However, when ...
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How to get LNC as a theorem using Frege's Prop Calculus?
So Im using axioms from,Frege propositional calculus and is there any way to derive Law of non contradiction as theorem from them.
The axioms
A → (B → A) | THEN-1
(A → (B → C)) → ((A → B) → (A → C)) ...
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