[0.] 0 is defined as ∅, as is usual in several main set theories wrought as foundations of mathematics.
[1.] x ∈ R
[2.] x ∈ ∅ → x ∈ ∅
[3.] As the statements on the two sides of the arrow are both false, statement 2. is vacuously true.
[4.] So ∅ ⊆ ∅ holds by vacuous truth. It also holds by the axiom of identity.
[5.] (x ∈ R) · (x ∈ ∅) → x ∈ R
[6.] The statement on the left side of the arrow is false and the statement on the right side is true (by 1.). Since a falsehood implies anything, statement 5. is true.
[7.] Statement 5. can be restated R ∪ ∅ = R
[8.] Which is to say, R ∪ ∅ = R is vacuously true.
[9.] Or, ∅ ⊂ R is vacuously true.
[10.] With 0 defined as ∅, we arrive at the notion that the statement 0 ∈ R seems to contain a vacuous truth.
[11.] We find that ∅ is both a member and a subset of R. Its subset status holds vacuously. Whether its membership status is also vacuous depends upon how membership is determined. If we say { } ∈ N and { } ⊂ N, the second statement holds vacuously, but the first is asserted as a definition and so is not vacuous. So 0's subset status is vacuous but its membership in the reals is real enough.
Comment: Here we have an interesting philosophical thought about the nature of 0. That is, it is questionable whether 0 really exists, though it is darned handy, and so it was necessary to find a way to make 0 subsist.
[1.] x ∈ R
[2.] x ∈ ∅ → x ∈ ∅
[3.] As the statements on the two sides of the arrow are both false, statement 2. is vacuously true.
[4.] So ∅ ⊆ ∅ holds by vacuous truth. It also holds by the axiom of identity.
[5.] (x ∈ R) · (x ∈ ∅) → x ∈ R
[6.] The statement on the left side of the arrow is false and the statement on the right side is true (by 1.). Since a falsehood implies anything, statement 5. is true.
[7.] Statement 5. can be restated R ∪ ∅ = R
[8.] Which is to say, R ∪ ∅ = R is vacuously true.
[9.] Or, ∅ ⊂ R is vacuously true.
[10.] With 0 defined as ∅, we arrive at the notion that the statement 0 ∈ R seems to contain a vacuous truth.
[11.] We find that ∅ is both a member and a subset of R. Its subset status holds vacuously. Whether its membership status is also vacuous depends upon how membership is determined. If we say { } ∈ N and { } ⊂ N, the second statement holds vacuously, but the first is asserted as a definition and so is not vacuous. So 0's subset status is vacuous but its membership in the reals is real enough.
Comment: Here we have an interesting philosophical thought about the nature of 0. That is, it is questionable whether 0 really exists, though it is darned handy, and so it was necessary to find a way to make 0 subsist.
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