The hyperreals exist only relative to a non-constructive ultrafilter whose existence depends on the Axiom of Choice, making their ontological status model-relative rather than absolute.

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Axiom of Choice(Foundations of mathematics; arises in second-order logic as the statement that every total binary relation has a choice function): Given a set A of non-empty pairwise disjoint sets, there exists a set B containing exactly one element from each set in A. When A is infinite, forming B requires making infinitely many simultaneous choices.
Non-constructive(metaphysics/logic): Something that exists or is true on its own, rather than being built up or created from other things.
Ontological status(in metaphysics (the study of what exists)): What kind of thing something is considered to be or how real it exists—for example, whether something is a physical object, a concept, a property, or something else entirely.
Ultrafilter(as used in set theory): A special mathematical object that picks out collections of things in a specific way; think of it as a very precise filter for organizing sets of numbers.
hyperreals(Nonstandard analysis): A number system developed by Robinson that extends the standard reals and satisfies the transfer principle with respect to first-order statements about the reals
model-relative(describing properties in logic and mathematics): Dependent on which particular version or interpretation of a system you're looking at; something can be true in one model but not in another.

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