A different interpretation for set theory
in [Logic]
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From: Nam Nguyen on 16 Jul 2010 00:17 The epsilon relation symbol ('e') is just a 2-ary predicate relation one, such as '<', ... This suggests that it's conceivable to have a different semantics for 'e', besides the set-semantics "being a member of". How about this semantics: x e y means either one of the followings: - the thought x originates from thought y, or equivalently - part of what thought y is about is thought x. We could replace "thought" by "perception", "concept", etc... In this semantics, e.g., the empty thought can be still be syntactically defined as ExAy[~(y e x)], just as the empty set. A few questions arise though: Q1. Would ZFC as a formal system "support" this semantics, in the sense that ZFC theorems could be as much about human thoughts, thinking, as it is about sets? Q2. What would "collection" mean in this semantics of "thoughts"? -- --------------------------------------------------- Time passes, there is no way we can hold it back. Why, then, do thoughts linger long after everything else is gone? Ryokan ---------------------------------------------------
From: David R Tribble on 16 Jul 2010 00:34 Nam Nguyen wrote: > In this semantics, e.g., the empty thought can be still be syntactically > defined as ExAy[~(y e x)], just as the empty set. > > Q2. What would "collection" mean in this semantics of "thoughts"? This would imply that all sets are collections of elements based on the ur-element of"empty thought", ie., all sets ate simply sets (and sets of sets) of empty thoughts.
From: Nam Nguyen on 16 Jul 2010 01:05 David R Tribble wrote: > Nam Nguyen wrote: >> In this semantics, e.g., the empty thought can be still be syntactically >> defined as ExAy[~(y e x)], just as the empty set. >> >> Q2. What would "collection" mean in this semantics of "thoughts"? > > This would imply that all sets are collections of elements > based on the ur-element of"empty thought", ie., all sets > ate simply sets (and sets of sets) of empty thoughts. Not quite quite sure if I follow. Suppose the _syntactical_ theory is still ZFC. Now in the new semantics we're supposed to only talk about "thoughts"; and no words "set", "collection" would be use. So instead of the canonical semantics "collection", what would be its counterpart semantics in this paradigm of "thought"? -- --------------------------------------------------- Time passes, there is no way we can hold it back. Why, then, do thoughts linger long after everything else is gone? Ryokan ---------------------------------------------------
From: Nam Nguyen on 16 Jul 2010 02:07 Nam Nguyen wrote: > David R Tribble wrote: >> Nam Nguyen wrote: >>> In this semantics, e.g., the empty thought can be still be syntactically >>> defined as ExAy[~(y e x)], just as the empty set. >>> >>> Q2. What would "collection" mean in this semantics of "thoughts"? >> >> This would imply that all sets are collections of elements >> based on the ur-element of"empty thought", ie., all sets >> ate simply sets (and sets of sets) of empty thoughts. > > Not quite quite sure if I follow. Suppose the _syntactical_ theory is > still ZFC. Now in the new semantics we're supposed to only talk about > "thoughts"; and no words "set", "collection" would be use. So instead > of the canonical semantics "collection", what would be its counterpart > semantics in this paradigm of "thought"? A closer look seems to suggest there are axioms that wouldn't make too much sense using the "thought" semantics, such as "Axiom of regularity" and "Axiom of infinity". Still that doesn't seem to mean we couldn't formulate another theory using L(ZF) for thought, I'd think. -- --------------------------------------------------- Time passes, there is no way we can hold it back. Why, then, do thoughts linger long after everything else is gone? Ryokan ---------------------------------------------------
From: Frederick Williams on 16 Jul 2010 06:58
Nam Nguyen wrote: > > The epsilon relation symbol ('e') is just a 2-ary predicate relation > one, such as '<', ... This suggests that it's conceivable to have a > different semantics for 'e', besides the set-semantics "being a member > of". > > How about this semantics: > > x e y means either one of the followings: > > - the thought x originates from thought y, or equivalently > - part of what thought y is about is thought x. "Part of" is axiomatized in mereology. -- I can't go on, I'll go on. |