Is Set Theory Based on Circular Reasoning?

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Discussion Overview

The discussion revolves around the foundational aspects of set theory and its relationship with first-order logic (FOL). Participants explore the implications of defining set theory in terms of FOL, particularly focusing on potential circular reasoning and the validity of proofs within this framework.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant questions whether set theory's reliance on first-order logic constitutes circular reasoning, as the definition of FOL presupposes a "domain of discourse" akin to a set.
  • Another participant agrees that this could be seen as circular reasoning, prompting further inquiry into the implications for the validity of proofs.
  • A different participant suggests that sets are basic undefined terms in mathematics, which can be characterized by their properties, thus sidestepping the circularity issue.
  • One participant proposes that while sets are primitive notions, the construction of sets is not explicitly defined, referencing Russell's paradox as an example of potential issues arising from this assumption.
  • Another participant reflects on the relationship between FOL and ZFC, suggesting that ZFC provides a rigorous framework for constructing well-formed sets, including natural numbers.

Areas of Agreement / Disagreement

Participants express differing views on whether the relationship between set theory and first-order logic is circular. Some agree that it raises concerns about proof validity, while others argue that sets as undefined terms mitigate this issue. The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

The discussion highlights limitations in the definitions and assumptions surrounding sets and domains of discourse, as well as the implications of these for foundational mathematics. Specific unresolved mathematical steps and the scope of definitions are noted but not resolved.

gop
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Hi

I'm reading some stuff about proof theory and set theory right now and one question comes to my mind.

Set theory is defined in terms of FOL (First Order Logic). Nevertheless, when we "define" first order logic we already have the notion of a "domain of discourse", which is basically the same as a set. We also can't say "everything" is the domain of discourse because then we would need a universal set in set theory which doesn't exist (at least not in ZFC)
But then, we are defining one thing in terms of the other without knowing what the other is.

Isn't that sort of circular reasoning?
 
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Yes it is.
 
well then the question would be isn't that a problem?

I mean how can we be sure that any proof is valid if we have to look at any quantifier and say okay that means "all x in the domain of discourse" but then we look up what it means that "x is in the domain of discourse" and we get another quantifier...
 
Sets are one of the basic undefined terms of mathematics. Anything fitting the characteristics of a set (contains objects not counting duplicates) can be considered a set. The same goes for point, line, etc.
 
Yes that would sound logical to just say okay that is a set the same way as we say okay this is a predicate and it has to be either true or false (in FOL).

Hmm, so we have the primitive notion of a set to model the domain of discourse but we don't really say how that set can be constructed (Thus, Russels paradox i.e. is not a problem because we assume that we already have a well defined set).

Then, when we have FOL we build ZFC (which enables a rigerous treatment of how to construct well-formed sets) which then in turn enables us to constructs sets like the natural numbers etc..

Is this correct?
 

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