A Galilean linear order is a structure G = hA, vi where A is an Aristotelian linear order and v is a binary relation, read “is no longer than” such that:
1. If a is a part of A then a v a. (This is to say that v is reflexive on the set of parts of A.)
2. If a and b are distinct parts of A then either a v b or b v a.
3. If a, b and c are distinct parts of A then a v b and b v c implies a v c. (This is to say that v is transitive on the set of parts of A.)
So v defines a non-strict linear order on the parts.
An order automorphism is a mapping from an ordered set to itself that preserves the order. That is, if f: S-->S is a one-to-one correspondence from S to itself then f is an order automorphism for < iff for all a,b in S, a < b implies f(a) < f(b)
Explain why there is only one order automorphism for the natural numbers N={1,2,3,...}
Assume there’s another order automorphism for the natural numbers N = {1,2,3…}, such as N1 = {2,1,3…}, then f(1) = 2, f(2) = 1, which means that in N1, f(1) > f(2), but 2 > 1, so this contradicts the given condition of if a < b implies f(a) < f(b). So, the assumption is wrong. Thus, there is only one order automorphism for the natural numbers N = {1,2,3…}.
What kinds of order automorphism are there for the integers Z={...,-2,-1,0,1,2,...} ?
An order automorphism is a mapping from an ordered set to itself that preserves the order. For example, the order automorphism W = { + or - (n-1) | n = 1, 2, 3…}, n being a natural number, is for the integers z = {... -2, -1, 0, 1, 2, …}
With reference to your answers, explain why we can give a unique structural description of 2 in the natural numbers (i.e., a sentence in terms of < that is true of 2 but no other element of N) but not in the integers.
The use of 2 or its multiples can express half of the natural numbers 2,4,6,8… where 1 < 2, 3,5,7…(2n-1). The structural place of 2 can be in two places in the integers because of the positive and negatives, but there is only one place of 2 in the positive natural numbers.
Explain why an affine transformation f(t) = gt + s preserves ratios of lengths of open intervals of real numbers (where the length of (a,b) is |b-a|). If absolute time is affine (i.e., if its structure is invariant under affine transformations when interpreted in terms of the reals), explain why it can be said to flow uniformly.
Since absolute time is affine, its transformation f(t) can be mapped into (a,b), of which the ratios of lengths are uniformly distributed. Because the ratio of lengths are uniformly distributed, this means that time flows uniformly.
(Newtonian absolute time is a Galilean linear order TA = hT , vi that is uniform with respect to itself under any order isomorphism of < that is an order isomorphism of v. There is no distinguished unit of time but ratios are preserved. Since distinct automorphisms describe translations and rescalings of T, we say that absolute time is affine and provides exactly the structure required to compare intervals of time along distinct motions.)
