MHB Understanding the Difference Between 1 and 2 in Natural Numbers

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The discussion centers on the mathematical distinction between the representations of the numbers 1 and 2 using the variable x in set notation. It emphasizes that if x represents a specific set, then both expressions xUx and xUxUx simplify to the same set, indicating no difference. However, if x is treated as a placeholder for any set, then the expressions can represent different unions of sets. The conversation also touches on the definition of sets, noting that two sets are considered equal only if they contain the same elements. Overall, the thread seeks clarity on the implications of notation in understanding natural numbers and set theory.
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first of all, id like to thank the admin for taking down a rather embarrassing post I had put up earlier. I'll try to do better this time.

N = any natural number
1 = xUx
2 = xUxUx

What exactly is the difference between 1 and 2? Is each instance of x different? If so, how?
 
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Someone just explained to me that sets that are not each other are different than each other because sets are different than what they are not

True or no?
 
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Simmer said:
Someone just explained to me that sets that are not each other are different than each other because sets are different than what they are not

True or no?
?
ANYTHING, whether a set or not, is "different than what they are not"! That's pretty much what "different" means.

I suspect that what they were trying to tell you is that a set is "defined" by what it contains. Two sets are "equal" if and only if they contain exactly the same things.
 
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Simmer said:
first of all, id like to thank the admin for taking down a rather embarrassing post I had put up earlier. I'll try to do better this time.

N = any natural number
1 = xUx
2 = xUxUx

What exactly is the difference between 1 and 2? Is each instance of x different? If so, how?
You wrote this- what do YOU intend "x" to mean? Normally, unless something is said to the contrary, one symbol corresponds to one mathematical object. In particular, if x represents one set then both xUx and xUxUx are the same, x, because the union of any set with itself is just itself again. xUx= x so xUxUx= (xUx)Ux= xUx= x.

However, if it is clearly stated that "x" is a "place holder", that "x" is not a specific set but just represents any set, then xUx might mean "the union of any two sets" and xUxUx might mean "the union of any three sets. (Though I would consider that poor notation.)

(What does "N is a natural number" have to do with this?)
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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