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## Main Question or Discussion Point

Hi, I joined recently and I'd like to start a do-discussion about the empty set. Why is it empty and why is there only one empty set?

It's "full of nothing", right?

Apparently the ontology of a singular but empty set is still a hurdle for the philosophers, yet mathematicians use this logical, infinitely 'variable' thing with 'nothing' in it blithely, or regularly. Is it the only regular thing in existence, since it contains the future or all future imaginary events? Does it, in fact, represent imagination?

from wikipedia: (my underlines)

Do we then imagine time itself, since all epistemological 'experience' initially is in {}?

And of course, we epistemologically know that the universe was in {}, and now there is exactly one (empty) universe, because it's full of "things" from the empty future that the full past asserts.

What do you think - do we require that there be exactly one {} (which is provable mathematically), so that it's always empty, since otherwise the universe would not have been asserted which, obviously it has been?

It's "full of nothing", right?

Apparently the ontology of a singular but empty set is still a hurdle for the philosophers, yet mathematicians use this logical, infinitely 'variable' thing with 'nothing' in it blithely, or regularly. Is it the only regular thing in existence, since it contains the future or all future imaginary events? Does it, in fact, represent imagination?

from wikipedia: (my underlines)

Assertion is therefore the 'realization' that the imagined (empty thing) in {}, is in fact real...?Axiomatic set theory

In Zermelo set theory, the existence of the empty set is assured by the axiom of empty set, and its uniqueness follows from the axiom ofextensionality. However, the axiom of empty set can be shown redundant in either of two ways:

* A logic such that provability and truth hold for both empty as well as nonempty domains is called afree logic. Set theory is almost never formulated with free logic as its background logic; hence many theorems of set theory are valid only if the domain of discourse isnonempty. Canonical axiomatic set theory assumes that everything in the (nonempty) domain is a set. Thereforeat least one set exists; call it A. By the axiom schema of separation (a theorem in some theories), the set B = {x | x∈A ∧ x ≠ x} exists and, having no members, is the empty set;

* The axiom of infinity, included in all mathematically interesting axiomatic set theories, not onlyassertsthe existence of an infinite set I (from which B in the preceding paragraph may be constructed), but typically requires that the empty set be a member of I.

Do we then imagine time itself, since all epistemological 'experience' initially is in {}?

And of course, we epistemologically know that the universe was in {}, and now there is exactly one (empty) universe, because it's full of "things" from the empty future that the full past asserts.

What do you think - do we require that there be exactly one {} (which is provable mathematically), so that it's always empty, since otherwise the universe would not have been asserted which, obviously it has been?

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