0 ≠ { }, or, 0 ≠ ∅
(I could ask an ungracious insinuating question at this point. 
)
Best to not! 
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I refrained. It was a terrible struggle, but I did.
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Well, sort of.
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It is actually a very standard part of foundations of mathematics:
0 = \{\}
1 = \{ 0 \} (which is the same as \{\{\}\})
2 = \{0, 1\}
and etc.
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I guess it depends on arbitrary definitions and axioms. 0 ≠ { } or 0 ≠ ∅ when 0 is an integer, which is the way Faizal was using it.
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I don’t think this one and the definition I supplied are mutually exclusive.
Neil, can you please clarify then… (and I am and always was terrible at math)…
The conversation above was regarding whether or not “zero” is “nothing.” Such that “something” was being shown to have come from “nothing.” Even in your example above, one can see that zero has two different meanings. In this case, it means that the set is empty (it has “nothing” in it) because there are zero items in the set:
0 = { }In this example, zero is “something” because the set that contains “zero” has one item in it:
1 = { 0 }But, furthermore and to the point at hand… an “=” in an equation signifies equality. So then, to use this as an example that something can be shown to come from nothing seems problematic:
… because if zero is “nothing” then “nothing equals nothing.” So, to complete the analogy from math as stated earlier, this would suggest that “nothing comes from nothing” or if zero is “something” then it would suggest that “something comes from something” right?
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I posted that, because it directly contradicts what @DaleCutler had posted.
Foundations of mathematics is a program of logic. And usually the foundationalists take set theory as the starting point. So the idea is to construct a model of arithmetic based on set theory.
It seems natural that if a natural number n is to be identified with a set, then it should be a set with n elements. So the number zero should therefore be the empty set.
Beyond that, it is just an inductive construction (or recursive construction). The natural number zero is taken to be the empty set. And, given a natural number n, it’s successor (or n+1) is taken to be the set formed by adding the new entity (the number n) to the set which constitutes n.
It amounts to saying that the natural number n is the set consisting of all smaller natural numbers. So
n = \{0, 1, \cdots , n-1 \}
Note, however, that this is all abstract formalisms. It doesn’t really tell us what we mean by “nothing” in ordinary life. And just because we can create all of mathematics starting with the empty set, it does not follow that the entire universe can be created from nothing. Mathematics doesn’t have any bearing on reality, except as a useful tool for studying it.
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Mathematics is itself a part of that reality, however.
No. +1 and +3 came from that zero, and neither is nothing (zero). Neither is -4 for that matter, perhaps.
Another way to look at it: If nothing, rather than something, came from nothing in that equation, then the claim “something cannot come from nothing” does not pertain to the universe, because the universe is also nothing.
This is just apologetic double-speak. Theistic arguments like the Kalam are constantly invoking laws of nature such as the conservation of mass/energy or the “law of causation” to support the claim that the universe was created by God. If the statement “laws are incapable of acting” is sufficient to dismiss an argument in which the universe is proposed to come into being thru processes that do not involve a god, then it should also refute those arguments in favour of God that rely on scientific laws.
What is a law if it doesn’t have something to act upon, something which demonstrates that it even is a law?!
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How so?
Great question! Who knows? However, if it is true that laws of nature do not apply when “nothing” exists, then there is no law of conservation of mass/energy to prevent an entire universe coming from nothing, is there?
It wouldn’t prevent it. But neither would it necessitate it, if by nothing you are actually referring to nothing as in “not anything.” If by nothing you mean in the sense of what physicists call nothing, then I’m not sure either of those statements would apply.
I am referring specifically to the “not anything” that theist apologists insist would exist if there was no god.
Assuming that such a state is even possible, you are saying that laws of nature do not apply in such a state. In which case, an entire universe could just come into being without a cause from this nothing, because there are no laws that prevent this.
Again neither would it be necessitated. And it’s easy to conceive of such in the imagination. Quite a different story to conceive of such in reality. But even if there is no material cause, that doesn’t rule out an efficient cause.
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Check with Stephen Hawking. He said that the law of gravity created the universe.