/sci/ - Science & Math

For one, it's non-effective in the context of some moderate other set theory axioms, as evidenced by the fact that it implies Excluded Middle for all propositions.

Consider [math] m [/math] rationals in the unit interval, for example, the equidistant m=5 values [math] [0,1/4,1/2,3/4,1] [/math].

I try to find the binary function that is (maybe in the metric sense) an optimal approximation to the multiplication on the rationals.

The issue is that if [math] x=a/A, y=b/B [/math], then need not be part of those m numbers.
That is to say, e.g. if a and b are integers in [math] [0,m-1] [/math] , then
[math] \frac {a}{m-1} \cdot \frac {b{m-1} = \frac {a\cdot b/(m-1)}{m-1} [/math]
where sadly [math] a \cdot b/(m-1) [/math] isn't necessary integers in the interval

One example in the literature for such an approximation is to replace the multiplication on the interval by the min-function.

This works for the boundaries:
[math] 0 \cdot a = min(0,a) [/math]
[math] 1 \cdot a = min(1,a) [/math]

but overestimates the values in-between, e.g.
1/2 * 1/4 = 1/8
while
min(1/2, 1/4) = 1/4

I'd like to get a function (can be a complicated polynomial, or anything descrete, whatever), that's overall metrically closer to multiplication.

Any ideas?

I need to work with the ring [math] \mathbb{Z}_2 \times \mathbb{Z}_2 [/math] and for computational purposes it would come in handy if it were isomorphic to a subring of [math] \mathbb{Z}_n [/math].

The ring [math] \mathbb{Z}_2 \times \mathbb{Z}_2 [/math] in the standard representation with elements
(0,0), (1,1), a=(1,0), b=(0,1)
has component-wise addition and multiplication and so (0,0)+x=x, (1,1)*x=x, x+x=0 e and then a+b=(1,1), a*a=a, b*b=b, a*b=0, etc.

If I'm not mistaken, I found that I can represent much of it with the set {0,1,3,4} and addition mod 2 and multiplication mod 6, e.g.

3*3 mod 6 = 9 mod 6 = 3
4 mod 6 = 9 mod 6 = 3
3*4 mod 6 = 12 mod 6 = 0

but noting plus 4 mod 2 can be 3, so it doesn't cut it. And indeed, I'd like to embed as subring of the ring Z_n, so the two mods shouldn't be different.

So is this possible? Is [math] \mathbb{Z}_2 \times \mathbb{Z}_2 [/math] the subring of Z_n for some n?

"all axioms" will depend on the language you want to talk about.

Given a set theory U, you can take any set X and consider the group G = (X, id_X) with one element (the identity). Of course, those groups are all isomorphic. Nevertheless, it means you have a group for each set, and thus the class of all such object doesn't form a set.
I suppose in a similar way, you could attempt to argue that for each set X, you can take the predicate P_X the uniquely characterizes X, and take e.g. "there exists and A such that P(A)" as an axiom. Then all such axioms alone would form a class and the corresponding category wouldn't by U-small.

If I needed to make sense of the question, really, I'd take it you speak of the syntactic category for a theory
https://ncatlab.org/nlab/show/syntactic+category

Any economics grad students here tried applying to work at the International Monetary Fund? How have your chances been?

https://youtu.be/DHhy2Gk_xik