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- logic - What is Naive Set Theory? - Philosophy Stack Exchange
You can see in SEP : Set theory and also The early development of set theory Alternative Axiomatic Set Theories The "standard" book is Paul Halmos, Naive Set Theory (1960) From Wikipedia : "Unlike axiomatic set theories, which are defined using a formal logic, naive set theory is defined informally, in natural language " But you must face the same problems; you need to introduce axioms in
- Is everything in the Naive Set Theory included in the Axiomatic Set . . .
The answer to "everything" is a trivial no Naive set theory is informal, and while its intuitions motivate axiomatic theories none of them can capture them all For one thing, intuitions are incoherent, so one has to choose what to capture, and informal theories mix base level and meta level arguments, while axiomatic theories can not do that
- How do philosophers of mathematics understand the difference between . . .
Short Answer It sounds you're struggling to understand the relationship between three fundamental theories Naive set theory is the theory used historically by Gottlob Frege to show that all mathematics reduces to logic Type theory was proposed and developed by Bertrand Russell and others to put a restriction on set theory to avoid Russell's paradox, and which was then replaced by ZF and ZFC
- philosophy of mathematics - Although Russells paradox has the virtue . . .
According to Tim Button the reason Russell's paradox is a problem in set theory is because set theory relies on classical first-order logic and one can express that paradox there First he considers the paradox from the perspective of naive set theory: (page 109) In part II, we worked with a naïve set theory But according to a very naïve conception, sets are just the extensions of
- philosophy of mathematics - Can any logic system provide the impossible . . .
In naive set theory in classical logic, we cannot describe or find a solution to Russell's set paradox (it is impossible) But is it there any logic system or any method that can provide this solution?
- Can one still derive paradoxes from the amended version of Naive Set . . .
The definition is different from definition of set in the naive set theory, which does lead to contradiction In other words, the sets in naive set theory are not sets in the sense the quote defines
- What framework or tool solves the Barber Paradox?
Rejecting naive set theory usually involves adopting an axiom system such as ZFC or NBG or NF We can reject classical logic, accept naive set theory and accept the paradoxical sentence as true This is the approach favoured by dialetheists such as Graham Priest It allows that we can make non-trivial statements about inconsistent objects
- How to write that two specific sets are the same set in first-order . . .
(1) The set whose only members are the prime numbers between 6 and 12 is the same as the set whose only members are the solutions to the equation x^2-18x+77=0 How do I translate statement (1) above to first-order naive set theory? The book I am following seems to define a set as follows: ∃a ∀x (x ∈ a ↔ P (x))
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