Loose Ends

Restated here is the example proof given in the Metaphysics section.

1) adding any positive integer to any other positive integer gives a sum strictly greater in magnitude.
2) Zero is less than every positive integer.
3) 0 + x = x for all positive integers x.

Conjecture: Zero is not a positive integer

Within this framework, we find that if we assume the opposite (the logical negative) of our conjecture, and apply it as one would an axiom, that zero is positive, then either;

(1) is false by means of (3)
or (2) by means of (1),
or if zero is positive, by (2) it is lower than itself, but by being positive, adding zero equates just that 0<0. However, this is not true in the strict sense by (1).

The axioms contradict the conjecture, so by contradiction, zero is not a positive integer.

We could also assume that zero is indeed positive, by means of modifying our axioms to exclude (1) as contradicted. Thus we have two very different systems, each with their own unique structure.

So, assuming that we are well versed in the usual operations of addition within the set of integers, we may state with certainty that 1 > 0. Then it may well be obvious that the statement ">=" to signify "greater than or equal to", immediiately entails the statement 1 >= 1.

We can also be certain on our own part of 0 >= 0.

However, if we modify our language of "addition" upon the integers with an ordering on the set to accommodate the statement that "zero is a positive integer", then by axiom 1, by adding zero, we arrive at a sum which is strictly greater in magnitude. Now, the predicate 'strictly greater' may either provide a contradiction, or we entail the property that 'strictly greater' by some feat is meant to mean '4 + 0 >= 4'.

So, upon limiting (1) above to simply 'greater' or '>' (by virtue not of altering the meaning of  the predicate 'strictly', but of the predicate 'positive' upon which 'strictly' depends for it's definition.) We begin:

An integer 'strictly greater' than another integer differs upon subtraction by the latter by a positive integer.

Of course, (1) is not alone, as (2) can collapse in like terms, since 0 >  0. hence 0 is less than every positive integer, and therefore as itself. and by induction of (3);  x > x for all integers x.

We have not altered our axioms, but have altered our set theory, which entails altered logic and new structure of our systems, all because of the application of the predicate 'positive'.

This system may appear nonsense, but it is clear from history that the concept of the integer 'zero' remained illusive for thousands of years. Could it be that simply, as we would have it in English, that 4 > 4 stated "4 increased by nothing." juxtaposed with  "4 is a constant" was once just as much a cause of antagonism as the two differing set-theories reversed? It took ages to formulate negative numbers, and longer for algebra to birth complex numbers, let alone sets like the Hamiltonians and Hyperreals.

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