Set Theory 2.0 – a first attempt

Definition of a Thing:

If a Thing is to exist, it must be, by necessity, at the very least ‘not’ that-from-which-it-exists. This ‘not’ is what enables it to exist, and as such is the transformation by which it exists. This transformation is the defining boundary of a Thing; the Thing is fully-bounded by “not”. But the existence of a boundary gives rise to there being two sides[1]. And so we understand that for a bounded Thing to exist, there must exist that Thing’s complement – that is, the that-from-which-it-exists. “not” is an involutory transformation in that a second “not” cancels both. However, in normal speech, this cancellation is referred to as “is” – where “is” is “not not”. So we understand that by being defined by ‘not’, an extant Thing is absolutely unique. If it was not, then it would not exist – because it would not not-be something else (it is not-not something else – thus it ‘is’ something else).

So our rule for the existence of a Thing might be phrased:

  • A Thing exists if and only if it is ‘not’ its complement, which is also a Thing.

or

  • A Thing cannot exist without its complement, which is also a Thing.

One may call into question whether the ‘that-from-which-ness’ is also a Thing in its own right. My answer to you is that in an infinite void of absolute sameness (anaximander’s “apeiron”), the existence of a fully-enclosing boundary is the only boundary of existence, and so fully encloses both ‘sides’. You can also attempt it from the other end of the spectrum and see that in the infinite and undifferentiated expanse of All, a boundary enclosing one side encloses the other also. Try it – don’t take my word for it. Another intuition pump is to imagine a circle around the word “this” written on a sphere. ‘outside’ that circle, on the other side of the sphere is the word “that”. Stretch the circle up over the equator and let it come to rest around the word “that”: both are enclosed by one boundary.

It is for this very reason that I qualify “not” as a transformation, for in transforming (or ‘deforming’ if you feel more comfortable with that word) a substance nothing is added or removed – it is merely changed. Indeed, form is formed by transformation.

Definition of a Set:

A “set” is intuitively understood to be a collection of Things. Far be it from me to deny this. However I will point out two facts:

  • A set is itself a Thing.
  • A set is the mental act of substituting a plurality for a singularity, at the cost of losing the distinction between the items ‘of’ a plurality. That is, to treat a collective as a singular Thing.

Indeed, we say “a” plurality, as though it were a single Thing. So that is what a set shall be – a single Thing.

A set, being a Thing then, is thus subject to the same principle of uniqueness and adjacency as all Things:

  • A Thing cannot exist without its complement; which is also a Thing

Respecting the initial intent of even conceiving of such a Thing as a set, we set the rule then that a set only exists when more than one Thing can be ‘grouped’ as a set. “Membership” to a set is a mistaken notion of Identity, because there is no ‘inside’ – only ‘beside’.

  • ‘Remove’ one Thing and you define a different pair of sets – a Subset and its complement.
  • ‘Add’ one Thing and you define the identity of a set that is the previous set and its complement.

Definition of ‘Sub’ sets:

Let S be the substitute Thing for a collection of adjacent and unique Things. Thus by the rule of existence, there is the complement of S, which we could call ¬S

Let A be the substitute Thing for a collection of adjacent and unique Things. Thus by the rule of existence, there is the compliment of A, ¬A.

A is a subset of S iff both A and ¬A are S

“But A and ¬A are ¬S” you might say. This is also true! S is only when A and ¬A are also. This is what I will call the Halting Condition of S.

Domain of Discourse:

A set S is the domain of discourse when it is the only Thing under discussion. The only way for this to be the case is when all subsets of S are ‘not’ ¬S . This creates however, a recursive definition with no halting condition. So the only way to stop at the definition of S is to say that all subsets of S are S and are ‘not’ ¬S. That is to say There is No Thing that is ¬S. And so one must ‘swap’ S for ¬S and thus define the subsets of S as being identical to ¬S. This cancels the recursion, and keeps all Things as subsets of S.

I can understand that this seems very contradictory – subsets of S being identical to S and ¬S… what nonsense! The only way I (in my limited capacity) can convey what I have in mind is to ‘cheat’ with language a bit by referring to ‘inside’ and ‘outside’ (even though I know this is a false notion!).

If you imagine S as a letter inside a circle, so that the circle is labelled “not”, then you can put S on either side so that whatever crosses the boundary of that circle ‘inherits’ the word “not”. Thus if you’re ‘standing inside the circle’, whatever is ‘inside’ is S and ‘outside’ is ¬S. If you’re standing ‘outside the circle’ whatever is ‘outside’ is S and what is ‘inside’ is ¬S. Yes? So to constrain the domain of discussion to whatever is ‘inside’ S and yet maintain the capacity to distinguish subsets of S, you must put S ‘outside’ (without the ¬) and thus you can sub-divide where you are (‘inside’) into A and ¬A and whatever other subsets you wish. S is ‘fixed’ ‘outside’ and there is nothing that is not S because:

  • You ‘know’ you’re ‘inside S’ and so all distinguished Things are ‘subsets of S’
  • Any attempt to step ‘outside’ results in you being ‘in S’ still.

You are thus perpetually ‘inside S’. That is to say you are limited in your discussion to S and only S.

Another way of imagining this is to imagine the boundary as very flexible – so flexible that no matter how big you stretch it, you can always stretch it some more. Like trying to imagine going beyond the boundary of the universe – no matter how far you go, because you are of the universe, your continued travel means the universe expands with you. Or you are the universe expanding itself. Yet, this still conforms to the rule of existence because S, being on the ‘outside’ is bounded by ‘not’, so exists as ‘not you‘. The subsets of S are not S – because you are not substituting them (losing the capacity to distinguish them) for S. You are keeping subsets of S beside S, simultaneously.

Oh what I would not give for a better language – or more skill with this one!

I am exhausted, and hungry. Forgive me but I must stop here for tonight. Please, if you have any insight, or disagreement, feel free to comment below!

“[…]a single body is inconceivable apart from a space in which it hangs. Definition, setting bounds, delineation – these are always acts of division and thus of duality, for as soon as a boundary is defined it has two sides.” Alan Watts The Way of Zen

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