05-22-2003, 12:27 PM
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#11
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Veteran Member
Join Date: May 2001
Location: US
Posts: 5,495
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Page's Axiom
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“All members X of any sets are matched to one or more axiomatic concepts; this is a requirement for membership. Recursively, for any axiomatic concept with an arbitrary label “X” we are free to choose the identity for X from the set of all representations.”
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and (using the symbol e to denote the relation "is a member of":
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Identity Confusion. One of the causes of Russell’s Antinomy is similar to that behind the Logical Strength paradox (see section 4.6) which shows that x = x is a contradiction when taken literally; we really mean either r1x = r2x, or rx = Rx. By the same token n e n is a misrepresentation; while it could be agreed that yes, n is n, to suppose that n contains things other than itself is to contradict the definition of n. So, a representation in the form n e n is more sensibly portrayed as rn e Rn, showing that representational n’s are ‘members’ of the represented n’s. Even this can be misleading, however, since it gives the impression that a single entity rn could occur multiple times (because by definition sets may contain more than one member). I suggest we understand that Rn is a collection of representational entities whose qualities are sufficiently similar to permit them the description “set”. The set “rn” is thus shorthand for {r1n, r2n…rnn}, however n itself must have been abstracted using an axiomatic description. Therefore, behind a set {r1n, r2n…rnn} there must be an ‘original’ set of unique represented items in the form {R1a, R2b …Rnn}. Through comparison with an axiomatic n, designated here as r0n, The act of recognition coerces a, b, c etc. to be conferred the identity “n”. So, this paragraph is a roundabout way of saying that membership of a set does not imply that the member is literally part of a set, merely the sharing of common characteristics with the other members of the set via an axiomatic concept contained within the mind. Creation of a set implies that the mind is quantifying the set’s possible members, abstracting the common features to make them eligible to be members if they pass testing against the axiom.
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...hence logics.
Cheers, John
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