a dark lord
Continuum hypothesis says that some of them might be not countable, but still not equal to the continuum.
More precisely, we seek a set X whose cardinality[/] is between the cardinality of the natural numbers and the (strictly bigger) cardinality of the real numbers.
Ah, I vaguely remember our lecturer reordering rational numbers, mapping them to ordinals. I see what you mean.
In my opinion objective truth is whatever is happening regardless of us observing, describing or understanding.
Let's sort out our terminology.
- The objective reality is what happens outside of our consciousness (i.e. everything except qualia), regardless of whether we observe it.
- The objective truth is a simplified model of the objective reality that reflects important (for us) features with an adequate (for us) accuracy.
- The criteria of objective truth is how we decide how close some hypothesis/theory is to the objective truth.
All the mathematics comes down to statements like "If [axioms] then [theorem]." It doesn't automatically make the axioms the objective truth, thus the theorems and the whole maths can't be called objective either.
Maybe the answer DOES depend on axioms. Maybe with some axioms there is a set between N and C, but with others there isn’t ?
The answer always depends on the axioms. It's enough to assume that ∀x, x≠x, then any equation will be false, and Fermat's too. I believe Peano axioms will remain consistent if we redefine this one, even though the whole system would be practically useless (not that mathematicians worry about such minor issues ).
besides being internally consistent
Internal consistency is an important point I missed.
So OK, we have mathematical patterns with internal consistency (which allows us discarding obviously implausible models). We have a mathematical apparatus that allows us bringing those models to a common basis.
So, what do you mean when saying that the material world is probably a result of mathematics? From a naive idealist point of view, an atom and a star both implement some common idea, but from the scientific point of view, you only need to describe an atom, and the star will be a necessary result of atom's properties.
Today's most fundamental area of physics is called quantum field theory, and most of the observable universe on most levels is a result of it. Is there really a place for Platon's idea?
Lazy programmers never document their code, making their successors suffer.That's something I never understood about God. (I understand everything about God except this one thing. )
Why does he want to keep his plans secret?