I attempted to explain in convoluted terms, a question I had before. The questions concerns, whether, there the cauchy field equation
1.F(xy)=F(x)F(y) and
2.F(x+y)=F(x)+F(y) with
3. F:[0,1]to [0,1], and F(1)=1;
max element of the literally just entail that F(x)=x,
Without further specifying the that F is monotoncally stricly increasing of continuous; Some claim that they just entail that the function just is continuous, but surely this must be just symptom of the real number system. Just using 1,2,3 can one actually derive that F(1/pi)=1/pi, as one can for rational number using 1, 3, and algebraic rationals (of which there are only countably many) using 1,2,3 is0.5=F(0.5)= F((1/sqrt2)^2)=[F(1/sqrt2)]^2; via 2. and so F(1/sqrt2)=1/sqrt(2).
If the field equations( 2), with (1) and (3),cannot capture precisely (or in a somewhat more precise sense, individuate certain transcendental values) even a singular transcendental value;
that is in some sense, over and above it being it approximated by rational numbers between it in say the ordinal ordering as i suppose they might be now approximated in ranked system, cauchy equation with F(3) could the same thing in a ranked (strictly monotoncally increasing) system, by itself I presume))
The only difference, is that the algebraic rationals may help with this as well, but one would not think it would make much difference.
The algebraic rationals, are still countable in number, just like the rationals
If this is correct, Then in what sense, do (1 and 2) they (if at all) establish continuity and linearity where F(x) or F(x)=0.
And if so, how is (2) generally established to begin with?
Using some independence methods, or assuming that the system is completely additive (uncountably so) or some such .?
With (1)
one can establish the rational homogeineity,and the functional additivity of irrational numbers, that are rational multiples of each other, but not their precise value unlike the case with (2) given (3).
With (2), (1) can capture the precise the value of rational numbers in an infinite system, in which cauchy equation's can be generalized to its full extent.
But not the precise value of certain algebraic rationals, whereas one can with (2) (1) and (3) for at least some of them.ie 1/sqrt2 can be read off the table, as it were.
But if its only the algebraic rationals (3) it can it capture in addition when combined with (1) and (2), then what difference it make, with regard to continuity. ?
That is, other than implying a few other conditions such as non-negativity and restricting the form of the linear equation, c=1, c=0, but does it establish that function just is F=cx with c\in {1,0} And if they establish
Unless it just gives you strict monotonicity of the function F:[0,1] to [0,1]
or rather monotonicity in some sense the strict version given the existence of some x not equal to zero, x>0, xneq 1 for which f(x) exists,and F(x)>0;
(1)F(x+y)=F(x)+F(y) (in complete generality) given in ranked infinite system with, (2)F(1)=1 and cauchy equations
and (3) F strictly monotone increasing on F:[0,1] to [0,1]
generally has this implication i think
(in place of, boundedness- measurablity; or continuity itself , or at a point, some other regularity /continuity like constraint)
Monotonicity clearly follows from additivity if you can prove
F(x)>0 for all x>0.
But this is not hard: if x>0, then there exists y such that
x=y*y
F(x) = F(y*y) = F(y)*F(y) >0.
So monotonicity follows from the basic properties. Further, elementary algebra shows that for all rationals r
F(r) = r
which together with monotonicity yields the result.
Note that this reasoning fails in the complex, and that discontinuous automorphisms do exist in that case.
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