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According to the SEP:

Another place to find applications of inconsistency in analysis is topology, where one readily observes the practice of cutting and pasting spaces being described as “identification” of one boundary with another.

For example, joining the segments [-1,0] & [0,1] at 0.

One can show that this can be described in an inconsistent theory in which the two boundaries are both identical and not identical, and it can be further argued that this is the most natural description of the practice.

Now points do not have boundaries, so we can't construct a line by 'joining' points, as one might imagine (the usual construction is to give (not join) a bare set of points and then add the right topology); but can we consider a limiting argument? That is:

Consider, in one picture, that the real line is made of line segments, say of unit size; and joined end-to-end; as in the example above; using inconsistent joins.

Then take the limit as the interval size goes to zero.

Does this work as a definition of the inconsistent real line?

  • I don't know if this would qualify as an answer because I don't know if I'm understanding correctly. The points of the joined set consist of [-1,0) and (0,1] together with the equivalence class {0,0} where these are considered to be two different 0's coming from two different spaces. So {0,0} is a point. This is not the same as [-1,1], but it is homeomorphic with the quotient topology. There isn't really any joining going on, we are constructing a new space. I don't understand how this is inconsistent. – Matt Samuel Jan 27 '15 at 15:35
  • @Samuel: This is right, but not what they are arguing for; the usual picture, the one you explicated, is one where no actual join is done - its a mirage; the 'natural' one is to simply consider the two end-points as 'merged' or 'joined'; and presumably this is what they means here; to do this in a classical logic, (I'm assuming here) can't be pushed through ie it becomes inconsistent; but according to the SEP, it is possible in some non-classical paraconsistent logic; I don't know the details of the theory; so would be interested in an account of it. – Mozibur Ullah Jan 27 '15 at 15:43
  • First you take the en.wikipedia.org/wiki/Disjoint_union_(topology) then you put an equivalence relation on the points and construct the en.wikipedia.org/wiki/Quotient_space_(topology) . – Matt Samuel Jan 27 '15 at 15:49
  • @MattSamuel: I do know those details :), I meant the inconsistent variety! – Mozibur Ullah Jan 27 '15 at 15:57
  • I see. Hence your question. Got it. – Matt Samuel Jan 27 '15 at 15:58

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