On 31 Jul 2011, at 14:15, Craig Weinberg wrote:

## Advertising

Reblogging myself here, but curious to see what you think of the idea that 1 cannot be proven greater than 0.

In which theory?

`The notion of proof is theory and definition dependent. (contrary to`

`computability, which is absolute, by Church thesis).`

`If you agree to define x < y by Ez(z+x = y) "E" = "It exists". I`

`assume classical logic + the axioms:`

x+0 = x x+s(y) = s(x+y)

`0 denotes the number zero, and s(x) denotes the successor of x, often`

`noted as x+1. Cf the whole theory I gave last week. I use only a`

`subset of that theory here.`

`So we have to prove that 0 < s(0). By the definition of "<" above, we`

`have to prove that Ez(z + 0 = s(0))`

But s(0) + 0 = s(0) by the axiom x + 0 = x given above.

`So Ez(0 + z = s(0)) is true, with z = s(0). (This is the usual use of`

`the existence rule of classical logic).`

`Of course we could have taken the theory with the unique axiom "1 is`

`greater than 0". For all proposition we can always find a theory which`

`proves it. The interesting thing consists in proving new fact in some`

`fixed theory, and change only a theory when it fails to prove a fact`

`for which we have compelling evidences.`

Bruno

Someone’s comment on the previous chart mentioned the difficulty (impossibility?) of proving that 1 > 0. It’s an interesting kernelthere, and it reminds me of the whole “time does not physicallyexist”realization. On one level, I can think of zero as having no different relation to 1 than it has with any other number. Zero does the same thing to any number as it does to one and should be thought of more properly as the hub of the decimal spiral.I’m no mathematician, but I suppose that 0 is also formally definedasan integer between 1 and -1 or something. Still it exposes the question of whether the elemental underpinnings of our ability to count is really anchored in anything at all other than our own anthropological conventions of counting. Beyond numbers themselves, it appears that the whole quantitative notion - of greater than or lessthan, and of ‘equal’ is nothing but a figment of our feelingsaboutorder. There may not be any inherent moreness to something than the absence of something. If it’s the same thing, it actually seems more palatable to see the absence of something being a condition predicated upon the things’ a priori presence, no? Even if we want to get into quantum atopoietic craziness where things come out of nothing, rendering such a possibility discretely seems to threaten the whole notion of mathematical coherence. If any or all quantities, variables, and formulas can be generated arbitrarily from0, then 0 would seem to be the same thing as ∞, and greater than 1orany other arithmetic expression. Anthrodeximal Numberline Maybe it’s time to create a new numberline, without all of the repetitive decimal numerals. Instead there could be a Wiki of new quantitative symbols and names which anyone can add to and own as a permanent vector in the schema. It would be easy to translate them to and from Arabic numerals online and some interesting possibilities for informal encryption and unanticipated mathematic-linguistic synchronicity. By removing the aspect of repetition, we would unmask the semantic bias of the math logos and arrive at a pure generic linear calibration defined only in it’s own idiosyncratic a-signifying terms. Sort of like breaking the mantra of math, it’s trance-like rhythms thatdisguise it’s human neurological origin from us. By adding moreuniquequalitative sense to the thing, the quality-flattening power drains out and the system seems to disqualify itself. --You received this message because you are subscribed to the GoogleGroups "Everything List" group.To post to this group, send email to everything-list@googlegroups.com.To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com.For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.

http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.