- show quoted text - "If I make a program that gives for each number n as input the output 1- (n%2) , I can be reasonably sure it's the desired sequence . The information is finite. You've got it backwards . We can use a finite amount of information to generate an infinite string . You could say we're never interested in the structure of the infinite string per se (in the sense that we never directly observe the infinite string in its entirety ), rather , in the structure and morphology of the different 'shapes of information' that describe the string .
Anyway, there are different 'shades' of mathematics along the line between 'finitism' and 'platonic realism'. What I said is only not realistic for some extreme forms of finitism . It works from 'effective computability' upwards. "
The information is finite if we have a program that outputs the given sequence. However, for the Universal Sentence 'Every natural number has a successor' the information content is clearly infinite.For we can have axiom systems where,for example 1 has no successor or 2 has no successor or 3 has no successor etc. It is this sentence ,along with the analogous sentences of Geometery 'Every line segment can be subdivided' And 'Every segment can be extended indefinitely ' That bring in the infinite into Maths and logic . That is what needs to be reconciled with reality. Consider the Turing Machine with the infinite tape. The very assumption of 'infinite tape' presupposes Infinite information, because verifying that the tape is Infinite would need infinite number of steps. Apoorv