Set-theoretical foundations of Mathematics with only bounded quantifiersIs there formal definition of universal quantification?The egg and the chickenWhat's a magical theorem in logic?Set-theoretical multiverse and foundationsFragments of Morse—Kelley set theoryVopenka's Principle for non-first-order logicsAre there fragments of set theory which are axiomatized with only bounded (restricted) quantifiers used in axioms?About the limitation by sizeHow much should the average mathematician know about foundations?Why aren't functions used predominantly as a model for mathematics instead of set theory etc.?
Set-theoretical foundations of Mathematics with only bounded quantifiers
Is there formal definition of universal quantification?The egg and the chickenWhat's a magical theorem in logic?Set-theoretical multiverse and foundationsFragments of Morse—Kelley set theoryVopenka's Principle for non-first-order logicsAre there fragments of set theory which are axiomatized with only bounded (restricted) quantifiers used in axioms?About the limitation by sizeHow much should the average mathematician know about foundations?Why aren't functions used predominantly as a model for mathematics instead of set theory etc.?
$begingroup$
It seems that outside of researchers in Mathematical Logic, mathematicians use almost exclusively bounded quantifiers instead of unbounded quantifiers. In fact, I haven't observed any other practice from the very first day on when I was a student.
For example, a logician would write
$forall a : ( a in mathbb R ) rightarrow ( a^2 geq 0 )$
whereas most working analysists and algebraists write
$forall a in mathbb R : a^2 geq 0$
On the other hand, most mathematicians I know accept the idea that all of mathematics can be built up from set-theoretical foundations alone (starting the natural numbers).
So there seems to be a set of assumptions, almost universally agreed upon, which most working mathematicians assume implicitly for their practice. These assumptions start with set theory but apparently exclude unbounded quantifiers. In fact, unless you attend a class in formal logic you might never encounter unbounded quantifiers.
It seems that most mathematicians use a subset of human language enhanced with a subset of mathematical language (avoiding universal quantifiers) as their working language.
Question: Have there been attempts at precisely identifying this mathematical sublanguage and the rules that it governs?
set-theory lo.logic mathematical-philosophy
asked 12 hours ago
shuhaloshuhalo
1,6411530
$endgroup$
add a comment |
$begingroup$
It seems that outside of researchers in Mathematical Logic, mathematicians use almost exclusively bounded quantifiers instead of unbounded quantifiers. In fact, I haven't observed any other practice from the very first day on when I was a student.
For example, a logician would write
$forall a : ( a in mathbb R ) rightarrow ( a^2 geq 0 )$
whereas most working analysists and algebraists write
$forall a in mathbb R : a^2 geq 0$
On the other hand, most mathematicians I know accept the idea that all of mathematics can be built up from set-theoretical foundations alone (starting the natural numbers).
So there seems to be a set of assumptions, almost universally agreed upon, which most working mathematicians assume implicitly for their practice. These assumptions start with set theory but apparently exclude unbounded quantifiers. In fact, unless you attend a class in formal logic you might never encounter unbounded quantifiers.
It seems that most mathematicians use a subset of human language enhanced with a subset of mathematical language (avoiding universal quantifiers) as their working language.
Question: Have there been attempts at precisely identifying this mathematical sublanguage and the rules that it governs?
set-theory lo.logic mathematical-philosophy
asked 12 hours ago
shuhaloshuhalo
1,6411530
$endgroup$
add a comment |
$begingroup$
It seems that outside of researchers in Mathematical Logic, mathematicians use almost exclusively bounded quantifiers instead of unbounded quantifiers. In fact, I haven't observed any other practice from the very first day on when I was a student.
For example, a logician would write
$forall a : ( a in mathbb R ) rightarrow ( a^2 geq 0 )$
whereas most working analysists and algebraists write
$forall a in mathbb R : a^2 geq 0$
On the other hand, most mathematicians I know accept the idea that all of mathematics can be built up from set-theoretical foundations alone (starting the natural numbers).
So there seems to be a set of assumptions, almost universally agreed upon, which most working mathematicians assume implicitly for their practice. These assumptions start with set theory but apparently exclude unbounded quantifiers. In fact, unless you attend a class in formal logic you might never encounter unbounded quantifiers.
It seems that most mathematicians use a subset of human language enhanced with a subset of mathematical language (avoiding universal quantifiers) as their working language.
Question: Have there been attempts at precisely identifying this mathematical sublanguage and the rules that it governs?
set-theory lo.logic mathematical-philosophy
asked 12 hours ago
shuhaloshuhalo
1,6411530
$endgroup$
It seems that outside of researchers in Mathematical Logic, mathematicians use almost exclusively bounded quantifiers instead of unbounded quantifiers. In fact, I haven't observed any other practice from the very first day on when I was a student.
For example, a logician would write
$forall a : ( a in mathbb R ) rightarrow ( a^2 geq 0 )$
whereas most working analysists and algebraists write
$forall a in mathbb R : a^2 geq 0$
On the other hand, most mathematicians I know accept the idea that all of mathematics can be built up from set-theoretical foundations alone (starting the natural numbers).
So there seems to be a set of assumptions, almost universally agreed upon, which most working mathematicians assume implicitly for their practice. These assumptions start with set theory but apparently exclude unbounded quantifiers. In fact, unless you attend a class in formal logic you might never encounter unbounded quantifiers.
It seems that most mathematicians use a subset of human language enhanced with a subset of mathematical language (avoiding universal quantifiers) as their working language.
Question: Have there been attempts at precisely identifying this mathematical sublanguage and the rules that it governs?
set-theory lo.logic mathematical-philosophy
set-theory lo.logic mathematical-philosophy
asked 12 hours ago
shuhaloshuhalo
1,6411530
asked 12 hours ago
shuhaloshuhalo
1,6411530
asked 12 hours ago
shuhaloshuhalo
1,6411530
asked 12 hours ago
shuhaloshuhalo
1,6411530
asked 12 hours ago
shuhaloshuhalo
1,6411530
1,6411530
add a comment |
add a comment |
1 Answer
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$begingroup$
The most ambitious (and well-argued) attempt at formulating a set-theoretical foundation of the type you are proposing using bounded quantification has been suggested by non other than Saunders Mac Lane in the last chapter of his book Mathematics, Form and Function.
Mac Lane dubbed his system ZBQC, which can be described as a weakening of Zermelo set theory in which the scheme of separation is limited to formulae with bounded quantification. Curiously, at the level of consistency strength, ZBQC is the only known (weak) upper bound to the consistency strength of Quine's system NF; moreover, it is known that that the urelement-version, NFU, of NF (in which the axiom of infinity is included) is equiconsistent wit ZBQC.
On the other hand, Adrian Mathias has critically-and-forcefully responded to Mac Lane's thesis to found mathematics on ZBQC; see here for an article of his addressed to philosophers and general mathematicians, and here for an article addressed to logicians.
answered 7 hours ago
Ali EnayatAli Enayat
10.6k13467
$endgroup$
1
$begingroup$
I"m not sure what "(weak) upper bound" means, but I'd understand "ZBQC is an upper bound to the consistency strength of NF" to mean that Con(ZBQC) implies Con(NF). Did you mean the converse of that?
$endgroup$
– Andreas Blass
3 hours ago
add a comment |
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1 Answer
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1 Answer
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$begingroup$
The most ambitious (and well-argued) attempt at formulating a set-theoretical foundation of the type you are proposing using bounded quantification has been suggested by non other than Saunders Mac Lane in the last chapter of his book Mathematics, Form and Function.
Mac Lane dubbed his system ZBQC, which can be described as a weakening of Zermelo set theory in which the scheme of separation is limited to formulae with bounded quantification. Curiously, at the level of consistency strength, ZBQC is the only known (weak) upper bound to the consistency strength of Quine's system NF; moreover, it is known that that the urelement-version, NFU, of NF (in which the axiom of infinity is included) is equiconsistent wit ZBQC.
On the other hand, Adrian Mathias has critically-and-forcefully responded to Mac Lane's thesis to found mathematics on ZBQC; see here for an article of his addressed to philosophers and general mathematicians, and here for an article addressed to logicians.
answered 7 hours ago
Ali EnayatAli Enayat
10.6k13467
$endgroup$
1
$begingroup$
I"m not sure what "(weak) upper bound" means, but I'd understand "ZBQC is an upper bound to the consistency strength of NF" to mean that Con(ZBQC) implies Con(NF). Did you mean the converse of that?
$endgroup$
– Andreas Blass
3 hours ago
add a comment |
$begingroup$
The most ambitious (and well-argued) attempt at formulating a set-theoretical foundation of the type you are proposing using bounded quantification has been suggested by non other than Saunders Mac Lane in the last chapter of his book Mathematics, Form and Function.
Mac Lane dubbed his system ZBQC, which can be described as a weakening of Zermelo set theory in which the scheme of separation is limited to formulae with bounded quantification. Curiously, at the level of consistency strength, ZBQC is the only known (weak) upper bound to the consistency strength of Quine's system NF; moreover, it is known that that the urelement-version, NFU, of NF (in which the axiom of infinity is included) is equiconsistent wit ZBQC.
On the other hand, Adrian Mathias has critically-and-forcefully responded to Mac Lane's thesis to found mathematics on ZBQC; see here for an article of his addressed to philosophers and general mathematicians, and here for an article addressed to logicians.
answered 7 hours ago
Ali EnayatAli Enayat
10.6k13467
$endgroup$
1
$begingroup$
I"m not sure what "(weak) upper bound" means, but I'd understand "ZBQC is an upper bound to the consistency strength of NF" to mean that Con(ZBQC) implies Con(NF). Did you mean the converse of that?
$endgroup$
– Andreas Blass
3 hours ago
add a comment |
$begingroup$
The most ambitious (and well-argued) attempt at formulating a set-theoretical foundation of the type you are proposing using bounded quantification has been suggested by non other than Saunders Mac Lane in the last chapter of his book Mathematics, Form and Function.
Mac Lane dubbed his system ZBQC, which can be described as a weakening of Zermelo set theory in which the scheme of separation is limited to formulae with bounded quantification. Curiously, at the level of consistency strength, ZBQC is the only known (weak) upper bound to the consistency strength of Quine's system NF; moreover, it is known that that the urelement-version, NFU, of NF (in which the axiom of infinity is included) is equiconsistent wit ZBQC.
On the other hand, Adrian Mathias has critically-and-forcefully responded to Mac Lane's thesis to found mathematics on ZBQC; see here for an article of his addressed to philosophers and general mathematicians, and here for an article addressed to logicians.
answered 7 hours ago
Ali EnayatAli Enayat
10.6k13467
$endgroup$
The most ambitious (and well-argued) attempt at formulating a set-theoretical foundation of the type you are proposing using bounded quantification has been suggested by non other than Saunders Mac Lane in the last chapter of his book Mathematics, Form and Function.
Mac Lane dubbed his system ZBQC, which can be described as a weakening of Zermelo set theory in which the scheme of separation is limited to formulae with bounded quantification. Curiously, at the level of consistency strength, ZBQC is the only known (weak) upper bound to the consistency strength of Quine's system NF; moreover, it is known that that the urelement-version, NFU, of NF (in which the axiom of infinity is included) is equiconsistent wit ZBQC.
On the other hand, Adrian Mathias has critically-and-forcefully responded to Mac Lane's thesis to found mathematics on ZBQC; see here for an article of his addressed to philosophers and general mathematicians, and here for an article addressed to logicians.
answered 7 hours ago
Ali EnayatAli Enayat
10.6k13467
edited 7 hours ago
answered 7 hours ago
Ali EnayatAli Enayat
10.6k13467
answered 7 hours ago
Ali EnayatAli Enayat
10.6k13467
answered 7 hours ago
Ali EnayatAli Enayat
10.6k13467
10.6k13467
1
$begingroup$
I"m not sure what "(weak) upper bound" means, but I'd understand "ZBQC is an upper bound to the consistency strength of NF" to mean that Con(ZBQC) implies Con(NF). Did you mean the converse of that?
$endgroup$
– Andreas Blass
3 hours ago
add a comment |
1
$begingroup$
I"m not sure what "(weak) upper bound" means, but I'd understand "ZBQC is an upper bound to the consistency strength of NF" to mean that Con(ZBQC) implies Con(NF). Did you mean the converse of that?
$endgroup$
– Andreas Blass
3 hours ago
1
1
$begingroup$
I"m not sure what "(weak) upper bound" means, but I'd understand "ZBQC is an upper bound to the consistency strength of NF" to mean that Con(ZBQC) implies Con(NF). Did you mean the converse of that?
$endgroup$
– Andreas Blass
3 hours ago
$begingroup$
I"m not sure what "(weak) upper bound" means, but I'd understand "ZBQC is an upper bound to the consistency strength of NF" to mean that Con(ZBQC) implies Con(NF). Did you mean the converse of that?
$endgroup$
– Andreas Blass
3 hours ago
add a comment |
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