/sci/ - Science & Math

True or false only 2 options

In the second case,it would be equivalent to false.
In the first one,if the sort the variables range over include the empty set (if it is defined on the powerset of a set),then why not.

>>4599634
>In the second case,it would be equivalent to false.
The question of true vs. false is a question of semantics. no statements in pure logic ask about the truth about a statement. All derived statements of logic are tautologies.

http://en.wikipedia.org/wiki/Formal_semantics_%28logic%29

In standard set theory (say ZF) you define the ordered pair in term of sets as

(a,b) := {{a},{a,b}}

http://en.wikipedia.org/wiki/Ordered_pair#Kuratowski_definition

So
(a,a) := {{a},{a,a}} = {{a},{a}} = {{a}}

In (Peano) Aritmetic you have 0 := ∅ and 1 := {0} = {∅}

http://en.wikipedia.org/wiki/Peano_Arithmetic#Set-theoretic_models

so

(∅,∅) = {{∅}} = {1}

Predicate logic uses sets informally, so there really is no empty set yet. (the empty set is not to be confused with 'absurdum'). So given these frameworks above as examples with the usual model of the pair, your first statement is isomorphic to the set which contains the number one and the second is the zero. No problem with that.

yes, a binary predicate that's always false
P=∅; no pairs in it.

>>4599747
You're a smart ass retard that like talking out of your ass, please never post here again.
Question of semantics and syntax are the same with Gödel's completeness theorem and soundness so fuck off with the formality when they don't matter.
The null set always exist. It's in the axiom of set theory and exist everywhere you go in math.
Also we practically never EVER care about how the underlining ordered pair structure is made in proofs, (a,b) could just as easily be {{a},{a,{b}}}. And we never care about the set theoretic construction of the natural numbers unless you want to write {0,1} lazy as 2.

>>4599827
>You're a smart ass retard that like talking out of your ass
Why do you insult me?
>Question of semantics and syntax are the same with Gödel's completeness theorem and soundness so fuck off with the formality when they don't matter.
OP was asking a question withing Predicate Logic and was using specific set. I pointing out to the second poster that 'true' or 'false' are relevant in semantics, not in pure logic. The purpose was to find out why he seems to associate the notion of an empty set with truth or falsehood. I don't see how you've cleared that up. Truth or false are relevant concepts when it comes to Gödels theorems, as you rightly say. "empty set", "no set", that has a priori not to do with falseness.

>The null set always exist. It's in the axiom of set theory and exist everywhere you go in math.
It exists in set theory true. In Zermelo–Fraenkel set theory it comes from the axiom of infinity
http://en.wikipedia.org/wiki/Axiom_of_infinity
The empty set is not a priori to be found in predicate logic.

>The null set always exist. It's in the axiom of set theory and exist everywhere you go in math.
Also we practically never EVER care about how the underlining ordered pair structure is made in proofs, (a,b) could just as easily be {{a},{a,{b}}}. >And we never care about the set theoretic construction of the natural numbers unless you want to write {0,1} lazy as 2.
I was just constructing models to show the use of the constructions in OPs post. This was the answer of the question. The answer is "yes, in a set theory context, these constructions are valid."

>>4599827
>You're a smart ass retard that like talking out of your ass
Why do you insult me?
>Question of semantics and syntax are the same with Gödel's completeness theorem and soundness so fuck off with the formality when they don't matter.
OP was asking a question within Predicate Logic and was using a specific set. I was pointing out to the second poster that 'true' or 'false' are relevant in semantics, not in pure logic. The purpose was to find out why he seems to associate the notion of an empty set with truth or falsehood. I don't see how you've cleared that up. Truth or false are relevant concepts when it comes to Gödels theorems, as you rightly say. "empty set", "no set", that has a priori not to do with falseness.

>The null set always exist. It's in the axiom of set theory and exist everywhere you go in math.
It exists in set theory - true. In Zermelo–Fraenkel set theory it comes from the axiom of infinity
The empty set is not a priori to be found in predicate logic.
http://en.wikipedia.org/wiki/Axiom_of_infinity

>The null set always exist. It's in the axiom of set theory and exist everywhere you go in math.
Also we practically never EVER care about how the underlining ordered pair structure is made in proofs, (a,b) could just as easily be {{a},{a,{b}}}. >And we never care about the set theoretic construction of the natural numbers unless you want to write {0,1} lazy as 2.
I was just constructing models to show the use of the constructions in OPs post. This was the answer of the question. The answer is "yes, in a set theory context, these constructions are valid."

>>4599747
Is she wearing scleral contact lenses, or am i just seeing things?

(repost due to false citation)

>>4599827
>You're a smart ass retard that like talking out of your ass
Why do you insult me?
>Question of semantics and syntax are the same with Gödel's completeness theorem and soundness so fuck off with the formality when they don't matter.
OP was asking a question within Predicate Logic and was using a specific set. I was pointing out to the second poster that 'true' or 'false' are relevant in semantics, not in pure logic. The purpose was to find out why he seems to associate the notion of an empty set with truth or falsehood. I don't see how you've cleared that up. Truth or false are relevant concepts when it comes to Gödels theorems, as you rightly say. "empty set", "no set", that has a priori not to do with falseness.

>The null set always exist. It's in the axiom of set theory and exist everywhere you go in math.
It exists in set theory - true. In Zermelo–Fraenkel set theory it comes from the axiom of infinity
The empty set is not a priori to be found in predicate logic.
http://en.wikipedia.org/wiki/Axiom_of_infinity

>Also we practically never EVER care about how the underlining ordered pair structure is made in proofs, (a,b) could just as easily be {{a},{a,{b}}}.
>And we never care about the set theoretic construction of the natural numbers unless you want to write {0,1} lazy as 2.
I was just constructing models to show the use of the constructions in OPs post. This was the answer of the question. The answer is "yes, in a set theory context, these constructions are valid."