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I have been considering the possibility of an approach to set theory that represents the following property of the free variable x, in the terms of parameters h and k: ((x = h) or (x = k))) by means of a set existence schema that uses something other than the approach that Frege used before he was informed of Russell's paradox.

It has come to my attention that in some quarters this may be received as being a sign of obsessive dedication to my own pet theory, given that Frege's approach is still being used in ZFC.

How important is it, if we are seeking true foundations for mathematics, to ensure that the property of x that restricts x to being equal to one of two named variables be represented exactly as Frege represented it?

To explain my train of thought, I am responding to the following comment:

"it's not clear what the motivation for the ZFC pairing axiom would be" Seriously? In what universe do you not want to be able to form pairs? Why does your pet principle take precedence over how sets are actually used? That's absolutely an example of something "going wrong" - what you have to throw out to accommodate it is far too much. –

The above comment was posted in response to the question ...

https://math.stackexchange.com/questions/3406295/what-goes-wrong-if-we-explain-russells-paradox-as-resulting-from-an-overly-rigi

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  • This is not a site for reviewing users' personal proposals. For proper consideration they should be submitted to professional literature. – Conifold Oct 27 '19 at 7:55
  • @Conifold --> Galois submitted at least two papers, and neither of them was published. Anything that I submitted would be almost certainly of lower quality and less clarity than what Galois submitted, so I am not confident that the reception of what I submitted would teach me anything useful. – Ren Eh Daycart Oct 27 '19 at 8:27
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    The peer review process was non-existent in Galois's time, and, be it as it may, the function of SE is not to provide feedback on personal explorations. "How important is it" is primarily opinion based, and hence off-topic here as well. – Conifold Oct 27 '19 at 8:46
  • @Conifold "the function of SE is not to provide feedback on personal explorations" --> Could you please identify the criteria that you apply to determine whether or not an SE question is a personal exploration? First impressions can mislead, but my first impression is that the following is a personal exploration: math.stackexchange.com/questions/3286548/… ? – Ren Eh Daycart Nov 1 '19 at 19:52
  • @Conifold: please note that the Tags for the linked question include the following: "recreational-mathematics", and that the question begins: "This is a little algorithm I made today, which may appear to be quite complex, so I will start with an example. Questions are at the end of the post." On first impression, it does appear to be a personal exploration, doesn't it? That is why it is very important for you to disclose information about the criteria for correctly classifying an item as either a personal exploration or not a personal exploration. – Ren Eh Daycart Nov 1 '19 at 20:13
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it's not clear what the motivation for the ZFC pairing axiom would be.

On the early development of set theory, it is very useful the SEP's entry dedicated to The Early Development of Set Theory.

Regarding the Pair axiom, see Zermelo’s Axiomatization of Set Theory : Zermelo's 1908 original monograph uses the Axiom of elementary sets, that asserts the existence of "basic" sets : nullset, singleton (for every object a), pair (for every a,b).

The principle is necessary in order to manufacture sets from the basic stuff (in Zermelo system : urelements) or from already existing sets. Iterating the basic construction, we can manufacture a set from three elements, and so on.

Regarding

the possibility of representing the following property of the free variable x, in the terms of parameters h and k: ((x = h) or (x = k))),

obviously in set theory the property defining the pair z={h,k} is represented by : ∀x (x ∈ z ↔ (x=h ∨ x=k)).

The issue is that, without the unrestricted Comprehension Axiom, we are not entitled to assert that, for every property expressible in the language, the corresponding set exists.

Comment : what do you mean by the following?

a set existence schema that uses something other than the approach that Frege used before he was informed of Russell's paradox.

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  • Frege used the approach of encoding a property of x via the convention that elements of a set correspond to values of x that satisfy P(x), but other conventions are possible. Please see the link provided below the words "To explain my train of thought [...]" that are in large, bold letters. – Ren Eh Daycart Oct 30 '19 at 1:01

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