Wednesday, April 10, 2024

Being entirely in different things at once

Sextus Empiricus in his Outlines, dealing with genus and its species, rips off an argument of the Parmenides. If we interpret the genus-species relation artificially in terms of ordinary geometric inclusion then obviously there will be problems.  But there is nothing contradictory about several objects a1,a2,...,an all sharing the same relation with the same object b, i.e. R(a1,c),R(a2,c),...,R(an,c). In fact they all share the same property λx.R(x,c)

But what if we question set theoretic equality and replace it with path-connectedness ? For instance xy means that for every zx we have that there is a continuous path pzw from z to w for some wy.  Then we can well have that x1,x2...,xny and at the same time yx1,...,yxn.

Consider the following set in the plane with the relative topology:

 Then by our definition we have that  {xi}{y} and {y}{xi} for i=1,...,5.

Also, what is a set of elements ? The elements must have a distinguishing property or relation prior to being organized into the set (this discussion needs to be greatly expanded).

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