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This basic idea is called the "Principle of Plenitude", and it is considered valid in realms like classical mathematics. Anything you can describe in a mathematically exact form, is assumed to be available for you to use mathematically.

But even then, blatant inconsistencies force you to choose some imaginary objects over others. The clearest traditional example is that you cannot have a single set of everything, and the ability to know what is in and outside of every set in the universe with perfect precision. Choosing both nets you Russel's Paradox: the set of all sets that do not contain themselves as elements must both include and exclude itself.

So you cannot have your cake and eat it, too, if you want to be able to navigate your imaginary universe in a way that preserves the ability to do things like use language, and map your imaginings onto the outside world occasionally.

Mathematics itself is therefore not uniquely determined, you can choose some concepts over others. But, aside from that latitude, which is not very broad, every 'plenary' imaginary universe that is internally stable is equivalent to some model of Mathematics as a whole, and every less complete one is equivalent to some mathematical model.

This basic idea is called the "Principle of Plenitude", and it is considered valid in realms like classical mathematics. Anything you can describe in a mathematically exact form, is assumed to be available for you to use mathematically.

But even then, blatant inconsistencies force you to choose some imaginary objects over others. The clearest traditional example is that you cannot have a single set of everything, and the ability to know what is in and outside of every set in the universe with perfect precision. Choosing both nets you Russel's Paradox: the set of all sets that do not contain themselves as elements must both include and exclude itself.

So you cannot have your cake and eat it, too, if you want to be able to navigate your imaginary universe in a way that preserves the ability to do things like use language, and map your imaginings onto the outside world occasionally.

This basic idea is called the "Principle of Plenitude", and it is considered valid in realms like classical mathematics. Anything you can describe in a mathematically exact form, is assumed to be available for you to use mathematically.

But even then, blatant inconsistencies force you to choose some imaginary objects over others. The clearest traditional example is that you cannot have a single set of everything, and the ability to know what is in and outside of every set in the universe with perfect precision. Choosing both nets you Russel's Paradox: the set of all sets that do not contain themselves as elements must both include and exclude itself.

So you cannot have your cake and eat it, too, if you want to be able to navigate your imaginary universe in a way that preserves the ability to do things like use language, and map your imaginings onto the outside world occasionally.

Mathematics itself is therefore not uniquely determined, you can choose some concepts over others. But, aside from that latitude, which is not very broad, every 'plenary' imaginary universe that is internally stable is equivalent to some model of Mathematics as a whole, and every less complete one is equivalent to some mathematical model.

Source Link
user9166
user9166

This basic idea is called the "Principle of Plenitude", and it is considered valid in realms like classical mathematics. Anything you can describe in a mathematically exact form, is assumed to be available for you to use mathematically.

But even then, blatant inconsistencies force you to choose some imaginary objects over others. The clearest traditional example is that you cannot have a single set of everything, and the ability to know what is in and outside of every set in the universe with perfect precision. Choosing both nets you Russel's Paradox: the set of all sets that do not contain themselves as elements must both include and exclude itself.

So you cannot have your cake and eat it, too, if you want to be able to navigate your imaginary universe in a way that preserves the ability to do things like use language, and map your imaginings onto the outside world occasionally.