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‎Let ‎$‎‎a$ ‎and ‎‎$‎‎b$ ‎are ‎two "‎objects". ‎What ‎is ‎the ‎meaning ‎of‎ ‎‎$a=b‎‎$‎? This is one of the deepest problems of philosophy and logic because one needs a complete information about "entity" of these two objects to compare them and entity is not a clear notion. There are many essentially different ways to define ‎‎$a=b‎‎$. Here we introduce some of them: ‎

Pseudo Definition (1): (Ontological notion of equality) We say that ‎‎two ‎objects ‎are "ontologically equal" if they consist from same "things".

Remark (1): The axiom of extensionality (AE) in set theory gives a formal definition of the "equality" notion in the sense of above point of view.

$AE:~~\forall x~\forall y~(x=y\longleftrightarrow \forall z~(z\in x\longleftrightarrow z\in y))$

Pseudo Definition (2): (Descriptive notion of equality) We say that ‎‎two ‎objects ‎are "descriptively equal" if they have same "properties".

Remark (2): We can formalize the above notion of equality using the comprehension axiom and the classic (and paradoxical) correspondence between sets and formulas (properties) as follows:

"Two sets are equal iff they satisfy the same formulas"

Note that by the incorrect classic view this is equivalent to:

"Two sets are equal iff they belong to same sets"

And the formalized version of above statement is "the axiom of descriptive equality":

$DE:~~\forall x~\forall y~(x=y\longleftrightarrow \forall z~(x\in z\longleftrightarrow y\in z))$

Now the question is:

Question: Can we change the notion of equality in mathematics from current "ontological concept" to the new "descriptive concept" without losing consistency? Precisely, is the following statement true?

$Con(ZFC)\Longleftrightarrow Con(ZFC - AE + DE)$

Achtung: The names which I used for equality notions in the above pseudo definitions are not standard. Many philosophers like Frege (in his famous paper: "On sense and reference") and Leibniz discussed these notions and they have their own German words to refer these concepts.

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Isn't this sort of like what the univalence axiom in type theory does? Your question is formulated in the framework of a material set theory (ZFC), but this sounds like the sort of question type theory is more suited to handling. (My knowledge of homotopy type theory is very sketchy, so this could be completely off base.) –  Daniel Hast Oct 6 '13 at 23:56
    
@Daniel: Interesting point. Please give me more explanation about what you mean in homotopy type theory. –  Ali Sadegh Daghighi Oct 7 '13 at 0:04
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I don't actually know all that much about homotopy type theory — your discussion of descriptive equality just reminded me of some things I've heard about the type-theoretic treatment of equality, namely, the way it formalizes the idea that "isomorphic objects satisfy all the same properties". Hopefully, someone who knows more about this can offer a better explanation. –  Daniel Hast Oct 7 '13 at 0:16
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Ali, as a piece of friendly advise, please do not post so many questions in a row. Many of them do not seem too well thought out. It may be better to invest more time in each question on your own, and only then posting them if still confused. But your last few questions are not really of appropriate level for the site. Please consider posting on math.stackexchange instead. If after a few questions it becomes clear that the level has improved so they will again be appropriate here, then of course post here again. –  Andres Caicedo Oct 7 '13 at 0:30
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@ Andres: Dear Andres. OK. I will be more careful in the next questions. Thanks for your exact editorial view in MathOverflow which keeps the research level of the website. –  Ali Sadegh Daghighi Oct 7 '13 at 6:27
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closed as off-topic by Andres Caicedo, Noah S, Andrey Rekalo, Carlo Beenakker, Boris Bukh Oct 7 '13 at 18:00

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2 Answers

First, you have written the axiom of extensionality incorrectly relative to the notion of set theory as a mathematical theory written as a theory written in first-order logic with identity. Compare your sentence to the statements in Jech or Kunen. The conditional direction of your AE is a consequence of the standard account of identity. That account invokes Leibniz' principle of indiscernibility of identicals. The usual axiom of extensionality is the converse, reverse conditional in your statement. You will find that both Jech and Kunen acknowledge this relationship to the theory of identity attached to first-order predicate logic.

Second, your DE is standard object identity. This statement refers to the notion as described in Frege's "Comments on Sense and Reference". To the extent that you associate this notion with properties, you associate sets with concepts. Apparently, this can be a source of contention in some circles. I surmise this from the remarks at FOM found in the link,

http://www.cs.nyu.edu/pipermail/fom/2013-June/017370.html

There are, apparently, certain idealistic perspectives on sets that are at odds with logicistic conceptualization.

In addition, there is the problem of a predicative ground for the iterative conception of set. In order for the notion of identity in your DE to be definite in the sense of first-order logic with identity, all subclasses of the universe which are sets must be "known". This will be problematic for some views on sets.

First-order model theory circumvents this relationship through use of a denotation schema. That is, for each pair consisting of a constant for the language and a variable of the language one has

'a' denotes x iff a=x

This should be compared with the more familiar

'Snow is white' is true iff snow is white.

By this means, the historical arguments for a predicative ground are preserved relative to the usual axiom of extensionality.

To be clear, the sentence you have labeled AE is the Fregean notion of concept identity. The paradigm of first-order logic with identity separates the indiscernibility of identicals from the identity of indiscernibles. It is only through this distinction that a set can be divorced from conceptual interpretation.

In reply to another question you asked I directed you to the link:

http://plato.stanford.edu/entries/identity-relative

in which Harry Deutsch includes the remark,

"...the false notion that in FOL= the identity symbol defines the relation I(A,x,y)."

This remark is located in section 5 of the article.

He defines the relation I(A,x,y) in section 1 of the article with the statements,

"Now let A be a set and define the relation I(A,x,y) as follows: For x and y in A, I(A,x,y) if and only if for each subset X of A, either x and y are both elements of X or neither is an element of X. This definition is equivalent to the more usual one identifying the identity relation on a set A with the set of ordered pairs of the form for x in A. The present definition proves more helpful in what follows."

It is common for mathematicians to think that identity is eliminable from the language of set theory because the syntax appears to allow it. However, there are ramifications involved because of its impact on the paradigm of first-order logic with identity. Deutsch' article at the Stanford Encyclopedia of Philosophy addresses your question precisely because the arguments for language-relative identity invoke the indiscernibility expressed in your DE.

A small text that can help clarify these matters for anyone interested is "Understanding Identity Statements" by Thomas Morris. Although he presents a theory of regulative identity that would not be of interest to most mathematicians, he precedes that presentation with three chapters on other roles played by identity statements. There is the objectual role that corresponds with the ontological interpretation. There is the metalinguistic role that corresponds with the semantic interpretation (this is the interpretation that treats informative identity such as 'x=y' as stipulative). And, there is the epistemic role that treats informative identity problematically. It is in this latter role where Leibniz' principles warrant the truth or falsity of an informative identity statement.

Well, I expect this reply to be ignored. But, many of your questions are pointing to very non-standard views. You should read Deutsch' article linked above. Also, read the article on Skolem's paradox at the Stanford Encyclopedia of Philosophy. You will find mention of papers by Skolem and Zermelo that you can find in "From Frege to Goedel" by van Heijenoort. It is important to understand the standard paradigm if you are going to entertain contrary views.

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Dear fom: There is nothing "wrong" about my axiom of extensionality. They are just trivially equivalent and we can assert the axiom in both shapes. For example look at the difference between asserting the axiom of power set, seperation, pairing,... in Kunen which seems weaker than these axioms in Jech but they are equivalent. –  Ali Sadegh Daghighi Oct 7 '13 at 8:03
    
@ali I did not say that your axiom was wrong. I said that it breaks the relationship between the standard view of how first-order logic with identity relates to set theory. None of the axioms you mention have anything to do with equality of sets in the same way that the axiom of extensionality. So the situation of those provably equivalent forms are irrelevant to my remarks. It is one thing to know that Leibniz had something to do with the principle of identity of indiscernibles. It is an entirely different matter to read Leibniz and ... –  user41002 Oct 7 '13 at 15:44
    
Tried turning a too-long non-answer in comments, but it doesn't work automatically... –  Scott Morrison Oct 8 '13 at 0:37
    
quoth fom: understand that his logic had been intensional; that the principle of identity of indiscernibles is fundamentally a principle of intensional logic; that one of Frege's contributions had been to argue against intensional logic on behalf of extensional semantics; and, that Tarski's semantic conception of truth is intended to correctly represent extensional semantics for formalized languages in the sense of Frege's admonitions against intensional logic in favor of a scientific language. –  Scott Morrison Oct 8 '13 at 0:37
    
quoth fom: What I did say is that you should understand the standard paradigm if you are going to pursue non-standard views. You are claiming something to be trivial on the basis of derivations in a logic with presuppositions which your syntactic alteration actually changes. The moderators should consider moving these questions to math.stackexchange.com –  Scott Morrison Oct 8 '13 at 0:37
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What you call DE is often taken as a definition of identity in set theory, and the axiom of extensionality can for that reason be taken as $\forall x(x \in y \leftrightarrow x \in z) \rightarrow \forall x(y \in x \leftrightarrow z\in x) $.

DE has a lore back to Leibniz, and Fraenkel et.al. in Foundations of Set Theory call the relation "congruence".

The definition of $x=y$ by means of $\forall x(y \in x \rightarrow z\in x)$ is adequate when a sufficient amount of non-predicative means are available in the theory.

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