Re: What if 1+1=3?
- From: alexey_r@xxxxxxx
- Date: 2 Nov 2005 01:43:20 -0800
Gene Ward Smith wrote:
> Wayne Throop wrote:
>
> > Obviously, one can "imagine" it, in the sense of imaginging the
> > sense impressions, or results of activities modeled by 1+1=3,
> > as in the apples discussed elsethread. But what it would *mean*
> > for the rules of arithmetic to be changed has never been made clear.
>
> If I try to add 1+1=3 to the axioms of Peano arithmetic, I run into the
> problem that 1+1 = S(1), and 3 = S(2) = S(1+1) = S(S(1)). But
> S(S(1))=S(1) directly violates the axioms. To get out of this mess, we
> need to get rid of some of the axioms.
>
> >From 1+1=3 I get 1=2 and therefore 0=1 by subtracting. Hence for any
> integer,
> n = n*1 = n*0 = 0. I conclude I can make axioms for 1+1=3 by making all
> of the integers equal to each other, and can model the situation with
> the set of integers being {0}, subject to the laws 0+0 = 0 and 0*0 = 0.
> By ridding myself of the nonequivalence of the successor of n with n, I
> get a consistent but not very interesting theory. In this theory,
> anything I write down of the form A+B=C or A*B=C will be true.
>
> To settle the vexing question of whether 3 is even because it is 1+1,
> or 2 is odd because it is 3, I need a theory of divisibility. I know
> that A divides B with quotient q and remainder r if q*A + r = B. But
> for any values of A, B, q, and r, this is now true. Hence, every number
> (and they are all the same) is both even *and* odd. I can also conclude
> all numbers are prime, because for a number N to be composite, there
> has to be a number M such that M is not equal to N or 1 and M divides N
> with no remainder. The second condition is easy, but the first is
> impossible since no number M differs from either N or 1.
>
> Now, I presume omnipotence entails that you can create a race of beings
> who think the above version of arithmetic is useful, and constantly
> apply it in their lives, with the whole thing working out with (a lot)
> of Divine help. But that clearly isn't what the people who are making
> the claim "God can make 1+1=3" mean. I contend they don't actually mean
> anything specific, unlike the scenario I just presented.
See Ted Chiang's "Division by Zero".
.
- References:
- Re: The moment the willing suspension of disbelief died
- From: John Schilling
- Re: The moment the willing suspension of disbelief died
- From: Wayne Throop
- What if 1+1=3?
- From: Gene Ward Smith
- Re: The moment the willing suspension of disbelief died
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