Is there any axiomatized theory of arithmetic that is much stronger to be afflicted by Gödel theorems? I've read that there are axiomatized theories that are weaker than the theorem's criteria, i.e not expressive enough, and their consistency is proved within the same theory. I wondered if there would be something like that, which is stronger than the Gödel theorem's criteria for a axiomatized theory.
I've just read about Cantor'd diagonal argument, and I have some questions about it...
Let's say we want to map every real number between 0 and 1 to natural numbers. If I'm not mistaken, that can be done this way:
If we have number of form 0.abcdef... (letters stand for decimals, but only some are shown since there is infinite amount of them), then we can produce number N which equals 2^a * 3^b * 5^c * 7^d * 11^e * 13^f * (next prime)^(next decimal).
For example, number 0.12 equals to 2^1 * 3^2 (* 5^0 * 7^0 * ...) = 18.
Given any natural number N, we can easily determine which real number it represents (if any). My first question is: is all this consistent with Cantor's diagonal argument? (Can both be true at the same time?)
Cantor proved there is no one-to-one mapping (not just any mapping), is that important for his result? If yes, it somehow seems intuitive to me, at least at the first sight, that one-to-one mapping can be achieved by simply removing natural numbers that don't represent any real...
I have been reading discussions on this site about the Principia and about Godel's incompleteness theorem. I would really like to understand what you guys are talking about; it seems endlessly fascinating. I was an English/history major, though, and avoided math whenever I could. Consequently I have never even taken a semester of calculus. The good news (from my perspective) is that I have nothing to do for the rest of my life except for working toward the fulfillment of this one goal I have: to plow through the literature of the Frankfurt School and make sense of it all. Understanding the methods and arguments of logicians would seem to provide a strong context for the worldview that inspired Horkheimer, Fromm, et al.
So yeah, where should I start? Do I need to get a book on the fundamentals of arithmetic? Algebra? Geometry? Or do books on elementary logic do a good job explaining the mathematics necessary for understanding the material?
As I said, I'm not looking for a quick solution. I...
Are physical and logical truths distinct and, if so, how are they related? Is one more fundamental than the other?
By ‘physical truth’ I mean something true in virtue of the laws of physics, such as ‘masses attract other masses’ (gravity) and by ‘logical truth’ I mean something true in virtue of logical or mathematical principles, like ‘2 + 2 = 4’.
Could there be a world where some of the physical truths of our world were false but all of the logical truths of our world were true? That is, a world where masses always repelled other masses but 2 + 2 = 4?
Conversely, could there be a world where some of the logical truths of our world were false but all of the physical truths of our world remained true? That is, a world where 2 + 2 = 5 but where, as in our world, masses attract other masses?
[We’ve been discussing this hours and feel in desperate need of professional guidance - please help!]