# Tagged Questions

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### 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 ...
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### Zermeloâ€“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 $1$ ...
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### 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 ...
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### 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 ...
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### I currently know Calculus I — What steps would I take to understand Zermeloâ€“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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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} \\ ...
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### 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?