Hello,
I’m wondering if someone would explain to the extent they care what are
some aspects of anti-foundational logics, where the Axiom of Regularity
or Foundation is relaxed as a requirement.
In an anti-foundational logic, there is no axiom that says a set can not
contain itself as a member element, in the context of sets. For
example, A = {A} is not a valid set in ZFC for reason of AR. So with a
different foundation axiom, completely different constructions might be
possible, for example special cases or a class of special cases.
What other of the axioms depend on the regularity axiom to validate
them?
The foundation axiom as it is contains some ambiguity in some
interpretations, basically the discussion here yielded the result that
not the set, but it’s transitive closure (set of all subsets and subsets
of subsets of the set) that is to be tested for regularity against the
actual axiom.
For example, it should be simple enough to list three reasons why the
Axiom of Foundation can be relaxed in a set theory otherwise compatible,
and three why not or that it would be meaningless. Yet, I do not
specifically do so now.
I’ve been reading about the lambda calculus and other predicate
calculi. Is there some kind of meta-grammar about the formulaic and
associative stuff like that? It uses, for example, variable assignation
and other computer programming metaphors and logical constructs.
Have a nice day,
Ross
—
Ross Andrew Finlayson
Finlayson Consulting
Ross at Tiki-Lounge: http://www.tiki-lounge.com/~raf/
"The best mathematician in the world is Maplev in Ontario." - Pertti L.
Ross A. Finlayson <rfinlay…@hot.rr.com> wrote in message
news:3A5DE21B.533B2D0D@hot.rr.com…
> Hello,
> I’m wondering if someone would explain to the extent they care what are
> some aspects of anti-foundational logics, where the Axiom of Regularity
> or Foundation is relaxed as a requirement.
> In an anti-foundational logic, there is no axiom that says a set can not
> contain itself as a member element, in the context of sets. For
> example, A = {A} is not a valid set in ZFC for reason of AR. So with a
> different foundation axiom, completely different constructions might be
> possible, for example special cases or a class of special cases.
> What other of the axioms depend on the regularity axiom to validate
> them?
None.
> The foundation axiom as it is contains some ambiguity in some
> interpretations, basically the discussion here yielded the result that
> not the set, but it’s transitive closure (set of all subsets and subsets
> of subsets of the set) that is to be tested for regularity against the
> actual axiom.
It doesn’t matter.
> For example, it should be simple enough to list three reasons why the
> Axiom of Foundation can be relaxed in a set theory otherwise compatible,
> and three why not or that it would be meaningless. Yet, I do not
> specifically do so now.
Reasons for: 1) it is convenient 2) most, if not all, structures arising in
mathematics can be shown to be isomorphic in some suitable sense
to a well-founded set. Reasons against: 1) it’s not really necessary;
if you drop it, all you have to do is to prefix the theorems using
the regularity axiom with condition that the sets mentioned are
founded.
–
Aatu Koskensilta (a…@mediaclick.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
On Thu, 11 Jan 2001 17:55:09 GMT,
aatu <zap…@sci.fi> wrote:
>> For example, it should be simple enough to list three reasons why the
>> Axiom of Foundation can be relaxed in a set theory otherwise compatible,
>> and three why not or that it would be meaningless. Yet, I do not
>> specifically do so now.
>Reasons for: 1) it is convenient 2) most, if not all, structures arising in
>mathematics can be shown to be isomorphic in some suitable sense
>to a well-founded set.
3) It’s easily shown to be consistent with ZF-Foundation; 4) you can
have nice cardinals for all sets in absence of AC. 5) building models
and proving things about them is much easier.
(all of these are elaborations of 1) of course)
–
Anatoly Vorobey,
mel…@pobox.com http://pobox.com/~mellon/
"Angels can fly because they take themselves lightly" – G.K.Chesterton
On 12 Jan 2001 10:50:46 GMT, Anatoly Vorobey <mel…@pobox.com> said:
> 4) you can have nice cardinals for all sets in absence of AC…
To what theorem are you alluding, and (if it is not evident from the
theorem) in what sense are these cardinals nice? Thanks.
Chris Menzel
Ross A. Finlayson <rfinlay…@hot.rr.com> wrote:
> [...]
> For example, it should be simple enough to list three reasons why the
> Axiom of Foundation can be relaxed in a set theory otherwise compatible,
> and three why not or that it would be meaningless. Yet, I do not
> specifically do so now.
In a book I read on nonstandard analysis, the author claimed that the
nonstandard models are easier to study if they are embedded in a
foundationless set theory. I lost the reference, but I’m pretty sure
it was post-1990. You can scan the literature…
–
Pierre Asselin
Westminster, Colorado
- Hide quoted text — Show quoted text -
Pierre Asselin wrote:
> Ross A. Finlayson <rfinlay…@hot.rr.com> wrote:
> > [...]
> > For example, it should be simple enough to list three reasons why the
> > Axiom of Foundation can be relaxed in a set theory otherwise compatible,
> > and three why not or that it would be meaningless. Yet, I do not
> > specifically do so now.
> In a book I read on nonstandard analysis, the author claimed that the
> nonstandard models are easier to study if they are embedded in a
> foundationless set theory. I lost the reference, but I’m pretty sure
> it was post-1990. You can scan the literature…
> —
> Pierre Asselin
> Westminster, Colorado
I am not surprised. An acronym of it might be ZF – R.
Ross
—
Ross Andrew Finlayson
Finlayson Consulting
Ross at Tiki-Lounge: http://www.tiki-lounge.com/~raf/
"The best mathematician in the world is Maplev in Ontario." - Pertti L.
On 12 Jan 2001, Chris Menzel wrote:
> On 12 Jan 2001 10:50:46 GMT, Anatoly Vorobey <mel…@pobox.com> said:
> > 4) you can have nice cardinals for all sets in absence of AC…
> To what theorem are you alluding, and (if it is not evident from the
> theorem) in what sense are these cardinals nice? Thanks.
I don’t know if this is what he meant, but you can get something
like cardinals without choice by defining the cardinality of a
set X to be the set of all sets Y such that
- the elements of Y can be put in a 1-1 correspondence
with the elements of X
- Y has minimal rank among the sets whose elements can be
put in a 1-1 correspondence with the elements of X
The cardinals are just those sets that are the cardinality of
something. This has properties like "X and Y can be put in a
1-1 correspondence with each other iff cardinality(X) =
cardinality(Y)". I don’t know what other "nice" properties they
may have …
–
Ian Sutherland email: isuth…@condor.depaul.edu
Sans Peur
On Fri, 12 Jan 2001 19:50:12 -0600, Ian Sutherland
<isuth…@condor.depaul.edu> said:
- Hide quoted text — Show quoted text -
> On 12 Jan 2001, Chris Menzel wrote:
> > On 12 Jan 2001 10:50:46 GMT, Anatoly Vorobey <mel…@pobox.com> said:
> > > 4) you can have nice cardinals for all sets in absence of AC…
> > To what theorem are you alluding, and (if it is not evident from the
> > theorem) in what sense are these cardinals nice? Thanks.
> I don’t know if this is what he meant, but you can get something
> like cardinals without choice by defining the cardinality of a
> set X to be the set of all sets Y such that
> – the elements of Y can be put in a 1-1 correspondence
> with the elements of X
> – Y has minimal rank among the sets whose elements can be
> put in a 1-1 correspondence with the elements of X
Yes, that is the usual way of defining the cards without choice.
But, as I understood him, Vorobey was alluding to a definition that
depended essentially on non-well-foundedness.
Chris Menzel
On 13 Jan 2001 02:37:07 GMT,
- Hide quoted text — Show quoted text -
Chris Menzel <cmen…@philebus.tamu.edu> wrote:
>On Fri, 12 Jan 2001 19:50:12 -0600, Ian Sutherland
><isuth…@condor.depaul.edu> said:
>> On 12 Jan 2001, Chris Menzel wrote:
>> > On 12 Jan 2001 10:50:46 GMT, Anatoly Vorobey <mel…@pobox.com> said:
>> > > 4) you can have nice cardinals for all sets in absence of AC…
>> > To what theorem are you alluding, and (if it is not evident from the
>> > theorem) in what sense are these cardinals nice? Thanks.
>> I don’t know if this is what he meant, but you can get something
>> like cardinals without choice by defining the cardinality of a
>> set X to be the set of all sets Y such that
>> – the elements of Y can be put in a 1-1 correspondence
>> with the elements of X
>> – Y has minimal rank among the sets whose elements can be
>> put in a 1-1 correspondence with the elements of X
>Yes, that is the usual way of defining the cards without choice.
>But, as I understood him, Vorobey was alluding to a definition that
>depended essentially on non-well-foundedness.
No, the minimal rank definition is what I had in mind. Sorry for
not spelling it out earlier.
–
Anatoly Vorobey,
mel…@pobox.com http://pobox.com/~mellon/
"Angels can fly because they take themselves lightly" – G.K.Chesterton
Anatoly Vorobey <mel…@pobox.com> wrote in message
news:slrn95toc6.us8.mellon@sasami.jurai.net…
> On Thu, 11 Jan 2001 17:55:09 GMT,
> aatu <zap…@sci.fi> wrote:
> >> For example, it should be simple enough to list three reasons why the
> >> Axiom of Foundation can be relaxed in a set theory otherwise
compatible,
> >> and three why not or that it would be meaningless. Yet, I do not
> >> specifically do so now.
> >Reasons for: 1) it is convenient 2) most, if not all, structures arising
in
> >mathematics can be shown to be isomorphic in some suitable sense
> >to a well-founded set.
> 3) It’s easily shown tconsistent with ZF-Foundationo be ;
Indeed almost trivial.
> 4) you can
> have nice cardinals for all sets in absence of AC.
Later on in this threat you elaborate on this and inform that you are
referring
to the Scott rank trick allowing us to use the "Russell-Frege style"
definition
for cardinals. How does this relate to non-foundational set theories,
however, escapes me. Perhaps I’m overlooking something here?
> 5) building models
> and proving things about them is much easier.
It’s nor all that hard in foundational set theory, provided you’re not
too pedantic about the details. Perhaps you could offer a good
example of a model that is intrisictly difficult to represent as a
founded set?
–
Aatu Koskensilta (a…@mediaclick.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
On Sat, 13 Jan 2001 16:52:58 GMT,
aatu <zap…@sci.fi> wrote:
>> >> For example, it should be simple enough to list three reasons why the
>> >> Axiom of Foundation can be relaxed in a set theory otherwise
>compatible,
>> >> and three why not or that it would be meaningless. Yet, I do not
>> >> specifically do so now.
>> >Reasons for: 1) it is convenient 2) most, if not all, structures arising
>in
>> >mathematics can be shown to be isomorphic in some suitable sense
>> >to a well-founded set.
These are your words. I understood you, here, to be listing reasons
for using the axiom of regularity, rather than reasons for dropping it.
>> 4) you can
>> have nice cardinals for all sets in absence of AC.
These are my words. I’m continuing the list of reasons for using
the axiom of regularity that you started. Therefore the following
question baffles me:
>How does this relate to non-foundational set theories,
>however, escapes me.
It doesn’t. It’s a (possible) reason to use foundation, not to drop it.
>> 5) building models
>> and proving things about them is much easier.
>It’s nor all that hard in foundational set theory,
I *am* talking about foundational set theory.
>too pedantic about the details. Perhaps you could offer a good
>example of a model that is intrisictly difficult to represent as a
>founded set?
I’m not aware of any. Someone else mentioned a (lost) reference
to a nonstandard analysis book, but I’m not familiar with it.
–
Anatoly Vorobey,
mel…@pobox.com http://pobox.com/~mellon/
"Angels can fly because they take themselves lightly" – G.K.Chesterton
Ross A. Finlayson wrote:
> For example, it should be simple enough to list three reasons why the
> Axiom of Foundation can be relaxed in a set theory otherwise compatible,
> and three why not or that it would be meaningless. Yet, I do not
> specifically do so now.
First, I should actually attempt to list these reasons.
There is probably an at least introductory treatment of this already.
First, why relaxing regularity is consistent and has meaning.
1) The other axioms of Zermelo-Fraenkel are consistent without it.
2) It allows anti-foundational sets consistently, which are useful.
3) Regularity is not completely well-connected to well-foundedness.
Second, why that is inconsistent or meaningless.
1) Some Cardinal operations employ foundation in sets, or rather,
well-foundedness, to maintain consistency.
2) Regularity maintains set characteristics that are useful in their own
right.
3) Ennui.
Ross
—
Ross Andrew Finlayson
Finlayson Consulting
Ross at Tiki-Lounge: http://www.tiki-lounge.com/~raf/
"The best mathematician in the world is Maplev in Ontario." - Pertti L.
Anatoly Vorobey <mel…@pobox.com> wrote in message
news:slrn9615uq.1lcv.mellon@sasami.jurai.net…
- Hide quoted text — Show quoted text -
> On Sat, 13 Jan 2001 16:52:58 GMT,
> aatu <zap…@sci.fi> wrote:
> >> >> For example, it should be simple enough to list three reasons why
the
> >> >> Axiom of Foundation can be relaxed in a set theory otherwise
> >compatible,
> >> >> and three why not or that it would be meaningless. Yet, I do not
> >> >> specifically do so now.
> >> >Reasons for: 1) it is convenient 2) most, if not all, structures
arising
> >in
> >> >mathematics can be shown to be isomorphic in some suitable sense
> >> >to a well-founded set.
> These are your words. I understood you, here, to be listing reasons
> for using the axiom of regularity, rather than reasons for dropping it.
My apologies. I were listing reasons to *use* the axiom of regularity,
not to drop it. This, however, is not apparent from the above quote.
Later in the same post I wrote:
"Reasons against: 1) it’s not really necessary;
if you drop it, all you have to do is to prefix the theorems using
the regularity axiom with condition that the sets mentioned are
founded.".
Which, by my standards, makes it clear that "against" refers to
"against using the axiom of regularity" and not "against dropping
the axiiom of regularity".
Of course, it is my responsibility to post sensibly worded posts,
and not yours to attempt to decipher the hidden meaning behind
ambiguously worded posts
> >> 4) you can
> >> have nice cardinals for all sets in absence of AC.
> These are my words. I’m continuing the list of reasons for using
> the axiom of regularity that you started. Therefore the following
> question baffles me:
> >How does this relate to non-foundational set theories,
> >however, escapes me.
> It doesn’t. It’s a (possible) reason to use foundation, not to drop it.
And as such makes perfect sense.
> >> 5) building models
> >> and proving things about them is much easier.
> >It’s nor all that hard in foundational set theory,
> I *am* talking about foundational set theory.
In which case you and I are in agreement.
> >too pedantic about the details. Perhaps you could offer a good
> >example of a model that is intrisictly difficult to represent as a
> >founded set?
> I’m not aware of any. Someone else mentioned a (lost) reference
> to a nonstandard analysis book, but I’m not familiar with it.
Neither am I. I don’t think non-standard analysis is a very hopeful
candidate in search for models that are difficult to interprete as sets
in foundational set theory.
Sorry for the confusion. My original post was rather ambiguous,
and I once again apologise.
–
Aatu Koskensilta (a…@mediaclick.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
- Hide quoted text — Show quoted text -
Pierre Asselin wrote:
> Ross A. Finlayson <rfinlay…@hot.rr.com> wrote:
> > [...]
> > For example, it should be simple enough to list three reasons why the
> > Axiom of Foundation can be relaxed in a set theory otherwise compatible,
> > and three why not or that it would be meaningless. Yet, I do not
> > specifically do so now.
> In a book I read on nonstandard analysis, the author claimed that the
> nonstandard models are easier to study if they are embedded in a
> foundationless set theory. I lost the reference, but I’m pretty sure
> it was post-1990. You can scan the literature…
> —
> Pierre Asselin
> Westminster, Colorado
Aczel describes for us a "Hyperset Theory", among others:
http://www.cs.man.ac.uk/~petera/LogicWeb/settheory.html
Ross
–
contact Ross
company Apex Internet Software
email i…@apexinternetsoftware.com
website http://www.apexinternetsoftware.com/
Ross <a…@calpha.com> writes:
>Pierre Asselin wrote:
>> Ross A. Finlayson <rfinlay…@hot.rr.com> wrote:
>> In a book I read on nonstandard analysis, the author claimed that the
>> nonstandard models are easier to study if they are embedded in a
>> foundationless set theory. I lost the reference, but I’m pretty sure
>> it was post-1990. You can scan the literature…
>> —
>> Pierre Asselin
>> Westminster, Colorado
>Aczel describes for us a "Hyperset Theory", among others:
>http://www.cs.man.ac.uk/~petera/LogicWeb/settheory.html
Here is another reference to a database application of non WF set
theory, and more.
Gerald Koenig
http://www.cs.bilkent.edu.tr/~akman/jour-papers/air/node10.html#SECTI...
- Hide quoted text — Show quoted text -
>Ross
>–
>contact Ross
>company Apex Internet Software
>email i…@apexinternetsoftware.com
>website http://www.apexinternetsoftware.com/
- Hide quoted text — Show quoted text -
Gerald Koenig wrote:
> Ross <a…@calpha.com> writes:
> >Pierre Asselin wrote:
> >> Ross A. Finlayson <rfinlay…@hot.rr.com> wrote:
> >> In a book I read on nonstandard analysis, the author claimed that the
> >> nonstandard models are easier to study if they are embedded in a
> >> foundationless set theory. I lost the reference, but I’m pretty sure
> >> it was post-1990. You can scan the literature…
> >> —
> >> Pierre Asselin
> >> Westminster, Colorado
> >Aczel describes for us a "Hyperset Theory", among others:
> >http://www.cs.man.ac.uk/~petera/LogicWeb/settheory.html
> Here is another reference to a database application of non WF set
> theory, and more.
> Gerald Koenig
> http://www.cs.bilkent.edu.tr/~akman/jour-papers/air/node10.html#SECTI...
> >Ross
> >–
> >contact Ross
> >company Apex Internet Software
> >email i…@apexinternetsoftware.com
> >website http://www.apexinternetsoftware.com/
So, there have been anti-foundational set theory concepts since there have been founded or
foundational set theory concepts. To preserve consistency in their current systems, many
adopted the foundational set theory concepts, which required the introduction of the class
concept as opposed to set, in set theory.
That’s an interesting paper. I’m reading the rest of it because it’s telling me some of what
these people are saying.
Ross
—
Ross Andrew Finlayson
Finlayson Consulting
Ross at Tiki-Lounge: http://www.tiki-lounge.com/~raf/
"The best mathematician in the world is Maplev in Ontario." - Pertti L.
"Ross A. Finlayson" <rfinlay…@hot.rr.com> writes:
xxx
- Hide quoted text — Show quoted text -
>Gerald Koenig wrote:
>> >Aczel describes for us a "Hyperset Theory", among others:
>> >http://www.cs.man.ac.uk/~petera/LogicWeb/settheory.html
>> Here is another reference to a database application of non WF set
>> theory, and more.
>> Gerald Koenig
>> http://www.cs.bilkent.edu.tr/~akman/jour-papers/air/node10.html#SECTI...
>=Ross
>So, there have been anti-foundational set theory concepts since there have been founded or
>foundational set theory concepts. To preserve consistency in their current systems, many
>adopted the foundational set theory concepts, which required the introduction of the class
>concept as opposed to set, in set theory.
Apparently a lot of the appeal of the ZF system is that it didn’t
disturb the existing system of mathematical or philosophical
structures. The sacrifice of urelements and self-referential
(circular) sets found in ordinary language was a price willingly
paid. OK, call me a set crank, but I feel these deletions in ZF
suck.
>That’s an interesting paper. I’m reading the rest of it because it’s telling me some of what
>these people are saying.
I find as an amateur that it is one of the clearest overviews I
have ever read. I am mainly interested in set theory translations
to ordinary language, both ways; constructing languages is a hobby
of mine. One topic of interest covered that interests me:
The use of a tense or sequence argument in the formation of WF, the
Well Founded sets:
Quote from site:
"With the preceding discussion of WF [well founded] the explanation
is not difficult. When we are forming a set z by choosing its
members, we do not yet have the object z, and hence cannot use it
as a member of z. The same reasoning shows that certain other sets
cannot be members of z. For example, suppose that < z is a member
of y > . Then we cannot form y until we have formed z. Hence y is
not available and therefore cannot be a member of z. Carrying this
analysis a bit further, we arrive at the following. Sets are formed
in “stages.” For each stage S, there are certain stages which
are before S. "
I have never before noticed an argument in set theory which invoked
time as an element. Stages of course involve time. The grammar of
time in ordinary language is tense. Most set reasoning is in the
"Necessarily true" modal, which means always true, or true at
whatever moment or time frame, past, present or future. Predications
which are necessarily true are modal in natural language, and are
a different animal entirely from an ordinary claim which is tensed
past, present, or future. Most set claims like phi(x), look like
ordinary predicate logic translations, such as dog(x), but they really
are equivalent to something like, "=(2+2,4)" which is always true.
The actual claim is, "Necessarily, 2+2=4"; which is a modal claim for
all time.
Offhand it seems to me the defining graphs of non-well-founded set
theory could be drawn from any starting point and thus NWF is not time
or sequence dependent for its structural formation.
Anyway non-well-founded set theory is a beautiful thing, and I hope this
thread isn’t dead, I have a lot to learn from it.
Jerry
- Hide quoted text — Show quoted text -
>Ross
>–
>Ross Andrew Finlayson
>Finlayson Consulting
>Ross at Tiki-Lounge: http://www.tiki-lounge.com/~raf/
>"The best mathematician in the world is Maplev in Ontario." - Pertti L.
- Hide quoted text — Show quoted text -
Gerald Koenig wrote:
> "Ross A. Finlayson" <rfinlay…@hot.rr.com> writes:
> xxx
> >Gerald Koenig wrote:
> >> >Aczel describes for us a "Hyperset Theory", among others:
> >> >http://www.cs.man.ac.uk/~petera/LogicWeb/settheory.html
> >> Here is another reference to a database application of non WF set
> >> theory, and more.
> >> Gerald Koenig
> >> http://www.cs.bilkent.edu.tr/~akman/jour-papers/air/node10.html#SECTI...
> >=Ross
> >So, there have been anti-foundational set theory concepts since there have been founded or
> >foundational set theory concepts. To preserve consistency in their current systems, many
> >adopted the foundational set theory concepts, which required the introduction of the class
> >concept as opposed to set, in set theory.
> Apparently a lot of the appeal of the ZF system is that it didn’t
> disturb the existing system of mathematical or philosophical
> structures. The sacrifice of urelements and self-referential
> (circular) sets found in ordinary language was a price willingly
> paid. OK, call me a set crank, but I feel these deletions in ZF
> suck.
I write some software, it reads and interprets a different computer language. The language has
two variants, one that is similar to a set, or set of sets, with some elements referring to
themselves, so it is not perhaps well-founded. The other variant is a complete programming
language where the elements are not predefined. So, in software logic there are non-well-founded,
or I have seen them termed illfounded or ill-founded, sets.
- Hide quoted text — Show quoted text -
> >That’s an interesting paper. I’m reading the rest of it because it’s telling me some of what
> >these people are saying.
> I find as an amateur that it is one of the clearest overviews I
> have ever read. I am mainly interested in set theory translations
> to ordinary language, both ways; constructing languages is a hobby
> of mine. One topic of interest covered that interests me:
> The use of a tense or sequence argument in the formation of WF, the
> Well Founded sets:
> Quote from site:
> "With the preceding discussion of WF [well founded] the explanation
> is not difficult. When we are forming a set z by choosing its
> members, we do not yet have the object z, and hence cannot use it
> as a member of z. The same reasoning shows that certain other sets
> cannot be members of z. For example, suppose that < z is a member
> of y > . Then we cannot form y until we have formed z. Hence y is
> not available and therefore cannot be a member of z. Carrying this
> analysis a bit further, we arrive at the following. Sets are formed
> in “stages.” For each stage S, there are certain stages which
> are before S. "
> I have never before noticed an argument in set theory which invoked
> time as an element. Stages of course involve time. The grammar of
> time in ordinary language is tense. Most set reasoning is in the
> "Necessarily true" modal, which means always true, or true at
> whatever moment or time frame, past, present or future. Predications
> which are necessarily true are modal in natural language, and are
> a different animal entirely from an ordinary claim which is tensed
> past, present, or future. Most set claims like phi(x), look like
> ordinary predicate logic translations, such as dog(x), but they really
> are equivalent to something like, "=(2+2,4)" which is always true.
> The actual claim is, "Necessarily, 2+2=4"; which is a modal claim for
> all time.
> Offhand it seems to me the defining graphs of non-well-founded set
> theory could be drawn from any starting point and thus NWF is not time
> or sequence dependent for its structural formation.
> Anyway non-well-founded set theory is a beautiful thing, and I hope this
> thread isn’t dead, I have a lot to learn from it.
> Jerry
> >Ross
> >–
> >Ross Andrew Finlayson
> >Finlayson Consulting
> >Ross at Tiki-Lounge: http://www.tiki-lounge.com/~raf/
> >"The best mathematician in the world is Maplev in Ontario." - Pertti L.
There are uses to each. Computationally, restricting sets to well-foundedness can make processing
efficient, that is more efficient than for the generic (ill-founded) set. Some set theories have
strictly restricted non-well-founded sets, so if that is the case for one, it might be necessary
to select a different.
Ross
—
Ross Andrew Finlayson
Finlayson Consulting
Ross at Tiki-Lounge: http://www.tiki-lounge.com/~raf/
"The best mathematician in the world is Maplev in Ontario." - Pertti L.
- Hide quoted text — Show quoted text -
Gerald Koenig wrote:
> "Ross A. Finlayson" <rfinlay…@hot.rr.com> writes:
> xxx
> >Gerald Koenig wrote:
> >> >Aczel describes for us a "Hyperset Theory", among others:
> >> >http://www.cs.man.ac.uk/~petera/LogicWeb/settheory.html
> >> Here is another reference to a database application of non WF set
> >> theory, and more.
> >> Gerald Koenig
> >> http://www.cs.bilkent.edu.tr/~akman/jour-papers/air/node10.html#SECTI...
> >=Ross
> >So, there have been anti-foundational set theory concepts since there have been founded or
> >foundational set theory concepts. To preserve consistency in their current systems, many
> >adopted the foundational set theory concepts, which required the introduction of the class
> >concept as opposed to set, in set theory.
> Apparently a lot of the appeal of the ZF system is that it didn’t
> disturb the existing system of mathematical or philosophical
> structures. The sacrifice of urelements and self-referential
> (circular) sets found in ordinary language was a price willingly
> paid. OK, call me a set crank, but I feel these deletions in ZF
> suck.
> >That’s an interesting paper. I’m reading the rest of it because it’s telling me some of what
> >these people are saying.
> I find as an amateur that it is one of the clearest overviews I
> have ever read. I am mainly interested in set theory translations
> to ordinary language, both ways; constructing languages is a hobby
> of mine. One topic of interest covered that interests me:
> The use of a tense or sequence argument in the formation of WF, the
> Well Founded sets:
> Quote from site:
> "With the preceding discussion of WF [well founded] the explanation
> is not difficult. When we are forming a set z by choosing its
> members, we do not yet have the object z, and hence cannot use it
> as a member of z. The same reasoning shows that certain other sets
> cannot be members of z. For example, suppose that < z is a member
> of y > . Then we cannot form y until we have formed z. Hence y is
> not available and therefore cannot be a member of z. Carrying this
> analysis a bit further, we arrive at the following. Sets are formed
> in “stages.” For each stage S, there are certain stages which
> are before S. "
> I have never before noticed an argument in set theory which invoked
> time as an element. Stages of course involve time. The grammar of
> time in ordinary language is tense. Most set reasoning is in the
> "Necessarily true" modal, which means always true, or true at
> whatever moment or time frame, past, present or future. Predications
> which are necessarily true are modal in natural language, and are
> a different animal entirely from an ordinary claim which is tensed
> past, present, or future. Most set claims like phi(x), look like
> ordinary predicate logic translations, such as dog(x), but they really
> are equivalent to something like, "=(2+2,4)" which is always true.
> The actual claim is, "Necessarily, 2+2=4"; which is a modal claim for
> all time.
> Offhand it seems to me the defining graphs of non-well-founded set
> theory could be drawn from any starting point and thus NWF is not time
> or sequence dependent for its structural formation.
> Anyway non-well-founded set theory is a beautiful thing, and I hope this
> thread isn’t dead, I have a lot to learn from it.
> Jerry
> >Ross
> >–
> >Ross Andrew Finlayson
> >Finlayson Consulting
> >Ross at Tiki-Lounge: http://www.tiki-lounge.com/~raf/
> >"The best mathematician in the world is Maplev in Ontario." - Pertti L.
Hi Gerald,
I have written some more about math that uses non-well-founded logic on news:sci.math.
Ross
–
contact Ross
company Apex Internet Software
email i…@apexinternetsoftware.com
website http://www.apexinternetsoftware.com/
In article <3A86169E.E634A…@calpha.com>, Ross <a…@calpha.com>
wrote:
> I have written some more about math that uses non-well-founded logic on
> news:sci.math.
> Ross
With general agreement on the quality of his logic.
Virgil wrote:
> In article <3A86169E.E634A…@calpha.com>, Ross <a…@calpha.com>
> wrote:
> > I have written some more about math that uses non-well-founded logic on
> > news:sci.math.
> > Ross
> With general agreement on the quality of his logic.
Capital, I say.
It’s not the quality of the logic, it’s the content, that is the part on
which not all agree.
Ross
–
Ross Andrew Finlayson
Finlayson Consulting, Est. 1994
Ross at tiki-lounge.com: http://neurosis.hungry.com/~raf/
"Have a nice day." FARS, DFARS, Berne Convention, USA rules may apply
On Sat, 10 Feb 2001 23:35:42 -0500, Ross <a…@calpha.com> wrote:
[...]
>Hi Gerald,
>I have written some more about math that uses non-well-founded logic on news:sci.math.
Not true. You have _said_ that you have done this, but nothing you’ve
written has _actually_ had _anything_ to do with the axiom of
foundation. Hard to decide whether you’re actually aware of this
or not – opinions differ on whether you’re a troll, an idiot or a
lunatic.
But in particular the axiom of foundation has nothing at all
to do with that simple proof that there is no bijection from
N to P(N). You seem to have the idea that just because
people talk about different flavors of set theory it follows
that anything anyone says is correct, if they just say
they’re using their own logic. That’s not the way it
works. If you had something interesting to say about
foundation that would be interesting, but when you say
that things you’ve written have something to do with
foundation that’s simply not so.
- Hide quoted text — Show quoted text -
>Ross
>–
>contact Ross
>company Apex Internet Software
>email i…@apexinternetsoftware.com
>website http://www.apexinternetsoftware.com/
On Sun, 11 Feb 2001 06:30:01 GMT, "Ross A. Finlayson"
- Hide quoted text — Show quoted text -
<rfinlay…@hot.rr.com> wrote:
>Virgil wrote:
>> In article <3A86169E.E634A…@calpha.com>, Ross <a…@calpha.com>
>> wrote:
>> > I have written some more about math that uses non-well-founded logic on
>> > news:sci.math.
>> > Ross
>> With general agreement on the quality of his logic.
>Capital, I say.
>It’s not the quality of the logic, it’s the content, that is the part on
>which not all agree.
No, there is general agreement that the logic is nonexistent and
the content is sheer gibberish. How you can say otherwise given
the unanimity of the replies to your posts is hard to understand –
could be you’re a troll, could be you’re too stupid or crazy to
see that everyone agrees you’re gibbering.
- Hide quoted text — Show quoted text -
>Ross
>–
>Ross Andrew Finlayson
>Finlayson Consulting, Est. 1994
>Ross at tiki-lounge.com: http://neurosis.hungry.com/~raf/
>"Have a nice day." FARS, DFARS, Berne Convention, USA rules may apply
"David C. Ullrich" wrote:
> On Sat, 10 Feb 2001 23:35:42 -0500, Ross <a…@calpha.com> wrote:
> [...]
> >Hi Gerald,
> >I have written some more about math that uses non-well-founded logic on news:sci.math.
> Not true. You have _said_ that you have done this, but nothing you’ve
> written has _actually_ had _anything_ to do with the axiom of
> foundation. Hard to decide whether you’re actually aware of this
> or not – opinions differ on whether you’re a troll, an idiot or a
> lunatic.
Here, I should note that Ullrich has a personal problem with me.
His focus is that he can call the police on me if I come ot visit his office in Oklahoma.
I wouldn’t do that, though, I would just go and sit in the math department, talking to the
students, secretaries, and professors. When Ullrich heard I was there, then he could lock
his door for his safety and call the campus police and tell them I was trespassing,
because I don’t have an Oklahoma State University student identification card. When the
campus police got there, if I didn’t bring them with me, they could ask me to leave, then
I could leave. I just won’t go and visit Ullrich.
Here Ullrich says, basically:
Ross -> Ross = troll or ross = idiot or Ross = lunatic.
Let’s call it Ullrich’s attempt at a Ross equation.
I have already told ullrich, and otherwise presented eveidence, that I am none of those
things. So, if Ullrich attempts to say "UAR", Ullrich’s Attempt at a Ross equation, I say
"Not".
> But in particular the axiom of foundation has nothing at all
> to do with that simple proof that there is no bijection from
> N to P(N). You seem to have the idea that just because
> people talk about different flavors of set theory it follows
> that anything anyone says is correct, if they just say
> they’re using their own logic.
No.
> That’s not the way it
> works. If you had something interesting to say about
> foundation that would be interesting, but when you say
> that things you’ve written have something to do with
> foundation that’s simply not so.
Ullrich would not get a failing grade from my class, he would have been ejected from my
class.
> >Ross
> >–
> >contact Ross
> >company Apex Internet Software
> >email i…@apexinternetsoftware.com
> >website http://www.apexinternetsoftware.com/
Ross
—
Ross Andrew Finlayson
Finlayson Consulting, Est. 1994
Ross at tiki-lounge.com: http://neurosis.hungry.com/~raf/
"Have a nice day." FARS, DFARS, Berne Convention, USA rules may apply