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There is a lot of writing both in favor and against AC from a philosophical standpoint - e.g. in favor see Penelope Maddy's Believing the axioms. However, there are also more mundane issues. I think that, whether or not it's ideal, a key point here is usability. An answer like this may seem dubiously appropriate at philosophy.stackexchange, but I think it'...


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ZFC is susceptible to Gödel's incompleteness as was the Principia Mathematica. See Gödel’s Incompleteness Theorems for an introduction to the theorem : Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor ...


2

It depends on what you mean by semantics, and by "work strictly with what equations or statements express." (In fact, I think this is vague enough that the question is borderline unanswerable; however, what follows may help clarify the issues.) The first point I want to make is that in my opinion natural language isn't an appropriate proxy. Natural language ...


1

In Peano arithmetic 2 is defined as the successor of 1 (in symbols : s(1)) and 1 in turn is s(0). Thus : 2=s(s(0)). In the same way : 4=s(3)=s(s(s(s(0)))). To prove the equation : 2+2=4 amounts to prove : s(s(0))+s(s(0)) = s(s(s(s(0)))). Repeated application of the axiom : n+s(m)=s(n+m) will produce the desired result. The "number ...


1

In truth-functional logic the semantic way to show validity is to use a truth table. The syntactic way would involve a derivation using rules of inference to go from any premises, line by line, to the conclusion. For example, suppose we want to show that the argument P, P → Q ∴ Q is valid. A Fitch-style natural deduction derivation would look like the ...


1

Neither Pythagoras nor a Pythagorean may have written the passage in question. One place to look for something like the passage would be Kenneth Sylvan Guthrie's The Complete Pythagoras which provides four surviving biographies of Pythagoras from antiquity and "a complete collection of the surviving fragments from the Pythagoreans". (page 1) Since Simone de ...


1

The famous 1964 paper of John Stewart Bell, in which "Bell's inequality" is established, begins by assuming two things. (The paper can be found here: https://cds.cern.ch/record/111654/files/vol1p195-200_001.pdf) One of them is "locality", which means (non-mathematically) that distant objects cannot affect one another instantaneously. Mathematically, Bell ...


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The eternal return not only requires infinite time but also a finite number of configurations that can take place during that infinite time. Wikipedia points this out by referencing Walter Kaufmann's quote of Heinrich Heine's earlier idea: Walter Kaufmann suggests that Nietzsche may have encountered this idea in the works of Heinrich Heine, who once ...


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