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first order logic written with set theoretic notation: Internal Set Theory
The Next CEO of Stack OverflowFirst-order vs. set-theoretic group theoryZFC set theory,first order theoryAre there axiomatizations of first order logic or set theory defined in first order logic or set theory?First Order Logic vs First Order TheoryFirst order logic and first order set theoryFirst-Order Logic into Set Theory?Lambda calculus combined with first order logic notation (quantifiers, propositional connectives, and set notation)Theory using First Order LogicQuestion about set theory and first order logicSet theoretic concepts in first order logic
$begingroup$
I've been reading Edward Nelson's 1977 paper on Internal Set Theory. In particular, I'm working through the proof that IST is a conservative extension of ZFC. In the preliminary results he sets up a operation $^*(-)$ which extends a set. This is just the ultrapower construction. This $^*(-)$ operation induces an isomorphism from the Boolean algebra of $P(V)$ into $P(^*V)$. Then there are a couple of results about how $^*(-)$ commutes with cartesian products (i.e. $(^*V)^n=^*(V^n)$. He then defines a projection $pi_j : P(V^n)to P(V^n-1)$ by
$$beginalignpi_j(E)=&langle x^1,...,x^j-1,x^j+1,...,x^nranglein V^n-1\ mid& mbox for some x^jmbox in Vmbox we have langle x^1,...,x^j-1,x^j,x^j+1,...,x^nranglein Eendalign$$
where $Ein P(V^n)$ which then induces a map on $P(^*V^n)to P(^*V^n-1)$. Furthermore $^*(-)$ commutes with these projection maps.
My question (thanks for reading along) is a remark he makes. "Logical connectives among relations, such as negation, implication, etc., are expressed by means of the projection operators $pi_j$ and universal quantifiers are expressed by a combination of them and complementation." Some of this makes sense to me. I'm confused by how to think about existential quantifiers in terms of these maps. I would love a reference or if you could provide me with a stepping stone to towards figuring this out. The pages in question are the bottom of 1193 and 1194. I've linked to the article below. Thank you.
Internal Set Theory: A New Approach to Nonstandard Analysis by Edward Nelson (Bull. Amer. Math. Soc, 83, No. 6, 1977)
EDIT: So for example, Let's look at $mathbbR^3$ and let $E=(x,y,z)inmathbbR^3 $. Then $pi_1(E)$ would be the unit disc in the z-y plane. How can connect this to something $exists w?in ?$
logic nonstandard-models alternative-set-theories
asked Mar 27 at 20:07
kevin robergekevin roberge
314
$endgroup$
add a comment |
$begingroup$
I've been reading Edward Nelson's 1977 paper on Internal Set Theory. In particular, I'm working through the proof that IST is a conservative extension of ZFC. In the preliminary results he sets up a operation $^*(-)$ which extends a set. This is just the ultrapower construction. This $^*(-)$ operation induces an isomorphism from the Boolean algebra of $P(V)$ into $P(^*V)$. Then there are a couple of results about how $^*(-)$ commutes with cartesian products (i.e. $(^*V)^n=^*(V^n)$. He then defines a projection $pi_j : P(V^n)to P(V^n-1)$ by
$$beginalignpi_j(E)=&langle x^1,...,x^j-1,x^j+1,...,x^nranglein V^n-1\ mid& mbox for some x^jmbox in Vmbox we have langle x^1,...,x^j-1,x^j,x^j+1,...,x^nranglein Eendalign$$
where $Ein P(V^n)$ which then induces a map on $P(^*V^n)to P(^*V^n-1)$. Furthermore $^*(-)$ commutes with these projection maps.
My question (thanks for reading along) is a remark he makes. "Logical connectives among relations, such as negation, implication, etc., are expressed by means of the projection operators $pi_j$ and universal quantifiers are expressed by a combination of them and complementation." Some of this makes sense to me. I'm confused by how to think about existential quantifiers in terms of these maps. I would love a reference or if you could provide me with a stepping stone to towards figuring this out. The pages in question are the bottom of 1193 and 1194. I've linked to the article below. Thank you.
Internal Set Theory: A New Approach to Nonstandard Analysis by Edward Nelson (Bull. Amer. Math. Soc, 83, No. 6, 1977)
EDIT: So for example, Let's look at $mathbbR^3$ and let $E=(x,y,z)inmathbbR^3 $. Then $pi_1(E)$ would be the unit disc in the z-y plane. How can connect this to something $exists w?in ?$
logic nonstandard-models alternative-set-theories
asked Mar 27 at 20:07
kevin robergekevin roberge
314
$endgroup$
add a comment |
$begingroup$
I've been reading Edward Nelson's 1977 paper on Internal Set Theory. In particular, I'm working through the proof that IST is a conservative extension of ZFC. In the preliminary results he sets up a operation $^*(-)$ which extends a set. This is just the ultrapower construction. This $^*(-)$ operation induces an isomorphism from the Boolean algebra of $P(V)$ into $P(^*V)$. Then there are a couple of results about how $^*(-)$ commutes with cartesian products (i.e. $(^*V)^n=^*(V^n)$. He then defines a projection $pi_j : P(V^n)to P(V^n-1)$ by
$$beginalignpi_j(E)=&langle x^1,...,x^j-1,x^j+1,...,x^nranglein V^n-1\ mid& mbox for some x^jmbox in Vmbox we have langle x^1,...,x^j-1,x^j,x^j+1,...,x^nranglein Eendalign$$
where $Ein P(V^n)$ which then induces a map on $P(^*V^n)to P(^*V^n-1)$. Furthermore $^*(-)$ commutes with these projection maps.
My question (thanks for reading along) is a remark he makes. "Logical connectives among relations, such as negation, implication, etc., are expressed by means of the projection operators $pi_j$ and universal quantifiers are expressed by a combination of them and complementation." Some of this makes sense to me. I'm confused by how to think about existential quantifiers in terms of these maps. I would love a reference or if you could provide me with a stepping stone to towards figuring this out. The pages in question are the bottom of 1193 and 1194. I've linked to the article below. Thank you.
Internal Set Theory: A New Approach to Nonstandard Analysis by Edward Nelson (Bull. Amer. Math. Soc, 83, No. 6, 1977)
EDIT: So for example, Let's look at $mathbbR^3$ and let $E=(x,y,z)inmathbbR^3 $. Then $pi_1(E)$ would be the unit disc in the z-y plane. How can connect this to something $exists w?in ?$
logic nonstandard-models alternative-set-theories
asked Mar 27 at 20:07
kevin robergekevin roberge
314
$endgroup$
I've been reading Edward Nelson's 1977 paper on Internal Set Theory. In particular, I'm working through the proof that IST is a conservative extension of ZFC. In the preliminary results he sets up a operation $^*(-)$ which extends a set. This is just the ultrapower construction. This $^*(-)$ operation induces an isomorphism from the Boolean algebra of $P(V)$ into $P(^*V)$. Then there are a couple of results about how $^*(-)$ commutes with cartesian products (i.e. $(^*V)^n=^*(V^n)$. He then defines a projection $pi_j : P(V^n)to P(V^n-1)$ by
$$beginalignpi_j(E)=&langle x^1,...,x^j-1,x^j+1,...,x^nranglein V^n-1\ mid& mbox for some x^jmbox in Vmbox we have langle x^1,...,x^j-1,x^j,x^j+1,...,x^nranglein Eendalign$$
where $Ein P(V^n)$ which then induces a map on $P(^*V^n)to P(^*V^n-1)$. Furthermore $^*(-)$ commutes with these projection maps.
My question (thanks for reading along) is a remark he makes. "Logical connectives among relations, such as negation, implication, etc., are expressed by means of the projection operators $pi_j$ and universal quantifiers are expressed by a combination of them and complementation." Some of this makes sense to me. I'm confused by how to think about existential quantifiers in terms of these maps. I would love a reference or if you could provide me with a stepping stone to towards figuring this out. The pages in question are the bottom of 1193 and 1194. I've linked to the article below. Thank you.
Internal Set Theory: A New Approach to Nonstandard Analysis by Edward Nelson (Bull. Amer. Math. Soc, 83, No. 6, 1977)
EDIT: So for example, Let's look at $mathbbR^3$ and let $E=(x,y,z)inmathbbR^3 $. Then $pi_1(E)$ would be the unit disc in the z-y plane. How can connect this to something $exists w?in ?$
logic nonstandard-models alternative-set-theories
logic nonstandard-models alternative-set-theories
asked Mar 27 at 20:07
kevin robergekevin roberge
314
asked Mar 27 at 20:07
kevin robergekevin roberge
314
edited 2 days ago
kevin roberge
asked Mar 27 at 20:07
kevin robergekevin roberge
314
asked Mar 27 at 20:07
kevin robergekevin roberge
314
asked Mar 27 at 20:07
kevin robergekevin roberge
314
314
add a comment |
add a comment |
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