0
votes
5answers
322 views

Are the real numbers really uncountable?

Consider the following statement Every real number must have a definition in order to be discussed. What this statement doesn't specify is how that loose-specific that definition is. Some examples ...
1
vote
3answers
70 views

Zermelo–Fraenkel set theory the natural numbers defines $1$ as $1 = \{\{\}\}$ but this does not seem right

If 1 can be defined as the set that contains only the empty set then what of sets which contain one thing such as the set of people who are me. number 1 does not just mean $1$ nothing, it means $1$ ...
0
votes
1answer
28 views

Evolution of Relations

In Frege, one finds relations treated as predicates in complex terms. However, modern set theory appears to treat them as two-place relation. Is this correct? If so, when did this shift occur and to ...
1
vote
1answer
68 views

When is it useful to reduce mathematical objects to foundational levels and when it is not?

When is it useful to reduce mathematical objects to foundational levels and when it is not? Let's say you work in the field of computer vision, or else. How can you claim your method is optimal if ...
2
votes
1answer
56 views

I currently know Calculus I — What steps would I take to understand Zermelo–Fraenkel set theory?

While this question can be discussed, it should have a clear answer by stating the following: How can one go from a high school / low-level college understanding of mathematics (completed Calculus ...
5
votes
7answers
256 views

What is the right interpretation of the axiom of extensionality

A set $a$ can be called extensional if it has the following propery: $$\forall b\left[\forall x\left[x\in b\iff x\in a\right]\Rightarrow a=b\right]$$ Based on this the axiom of extensionality can be ...
7
votes
5answers
2k views

how do we assume there is infinity?

Definition of infinite: A set is infinite iff it is equivalent to one of its proper subsets. We know that our universe doesn't contain infinite number of elements, so how do we assume there is ...
9
votes
1answer
440 views

Why is CH true if it cannot be proved?

Continuum hypothesis (CH) states that there can be no set whose cardinality is strictly between that of integers and real numbers. Godel, 1940 and Paul Cohen,1963 showed that CH can neither be proved ...
4
votes
3answers
200 views

Yablo's paradox? a paradox without self-reference [closed]

Yablo's paradox arises from considering the following infinite set of sentences: $$(S_1): \mbox{for all }k > 1, S_k\mbox{ is false} \\ (S_2): \mbox{for all }k > 2, S_k\mbox{ is false} \\ ...
20
votes
5answers
644 views

What does it mean for a set to exist?

Is there a precise meaning of the word 'exist', what does it mean for a set to exist? And what does it mean for a set to 'not exist' ? And what is a set, what is the precise definition of a set?
12
votes
1answer
493 views

Is First Order Logic (FOL) the only fundamental logic?

I'm far from being an expert in the field of mathematical logic, but I've been reading about the academic work invested in the foundations of mathematics, both in a historical and objetive sense; and ...
6
votes
6answers
943 views

Inherently discrete concepts

Are there any concepts which are naturally defined only for the integers and so far has resisted any attempts at extension to other fields such as rationals or reals? Does not meet criteria: ...