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This is a bit philosophical.

What does it mean to say that a mathematical object exists?

To add some concrete thoughts, I recently read the following:

"The empty set has the property that for all objects x, the statement 'x is in the empty set' is false."

But this statement reeks of all sorts of logical complexities to me. For one, x is left completely undefined and ambiguous. Typically, the phrase "for all..." in mathematics is used to say that every member of a certain set has a certain property. But in that case, which set does x belong to? It's well-known that there is no "universal set" of all mathematical objects in standard ZF set theory. So if x is not specified as belonging to a specific set, and yet it cannot belong to some "universal set" becase such a thing is contradictory under the standard ZF axioms, in what way does x meaningfully exist? And furthermore, how is the quoted statement above meaningful at all when the primary object of interest, x, cannot be established to even meaningfully exist?

Now, as far as I can tell, the quoted statement could be reworded to produce something that's not so logically pernicious. For example, this seems an improvement: "Suppose A is any non-empty set. Then for each member x of A, the statement 'x is in the empty set' is false." But still, the quoted statement raises some concerns about the exact meaning of mathematical existence.

What does it mean to say that a mathematical object exists?

To add some concrete thoughts, I recently read the following:

"The empty set has the property that for all objects x, the statement 'x is in the empty set' is false."

But this statement reeks of all sorts of logical complexities to me. For one, x is left completely undefined and ambiguous. Typically, the phrase "for all..." in mathematics is used to say that every member of a certain set has a certain property. But in that case, which set does x belong to? It's well-known that there is no "universal set" of all mathematical objects in standard ZF set theory. So if x is not specified as belonging to a specific set, and yet it cannot belong to some "universal set" becase such a thing is contradictory under the standard ZF axioms, in what way does x meaningfully exist? And furthermore, how is the quoted statement above meaningful at all when the primary object of interest, x, cannot be established to even meaningfully exist?

Now, as far as I can tell, the quoted statement could be reworded to produce something that's not so logically pernicious. For example, this seems an improvement: "Suppose A is any non-empty set. Then for each member x of A, the statement 'x is in the empty set' is false." But still, the quoted statement raises some concerns about the exact meaning of mathematical existence.

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