Infinity and vacuous names

Rather than just picking holes in Speranza's posts on infinity I thought I should say something more constructive.

Speranza raised the question how one would add an axiom of infinity into Grice's formal treatments relating to vacuous names, so I had a look back at my rehash of this material to see how that might work.

Before coming to Grice its worth pointing out some different kinds of axiom of infinity.

Speranza discussed infinity in connection with "finitism", and the definition of a finitist (at Wikipedia) is "only accepts the existence of finite mathematical objects".
 This means that the axiom of infinity in NBG which Speranza discussed would not be acceptable to a finitist.

"The" axiom of infinity also arises in the context of Russell and Whitehead's Principia Mathematica.
To do arithematic in Russell's Theory of Types you need an axiom of infinity, and it is a curiosity that Russell remained a logicist even though he considered the axiom of infinity used in Principia to be contingent.
The need of an axiom of infinity (or something equivalent) for doing arithmetic, is not peculiar to Russell's Theory of Types, is it common to all treatments of arithmetic including those approved by finitists (such as PRA), so how can this be if finitists reject the axiom of infinity?

The answer is that there are two quite different things which an axiom of infinity might do.
In the case of ZFC, NBG and most set theories, the axiom of infinity asserts the existence of a set with infinitely many elements.
To do arithemetic we do need infinitely many numbers, but  we don't necessarily have to have them collected together into a set.
In PA and in PRA, and in most first order formalisations of arithmetic, you have an axiom of infinity or its equivalent, but you don't have any infinite objects.
In fact the Peano axioms can be read as a way of asserting that there are infintely many things.

These give us another general formula for asserting that there are infinitely many things which does not require the existence of sets.
We assert the existence of a distinguished element (think zero) and a one-one function (thing successor) such that the distinguished element is not in the range of the function (zero is not a successor).
This ensures that if as we count from the distinguished element using the function to increment, we get a new object every time and count our way through an infinity of objects (numbers).
[sorry, it does here look like we have asserted the existence of a sucessor function, which itself is an infinite object, but though we make use of it we don't actually assert its existence, and it isn't in the range of our quantifiers, so there is a bit of (logically coherent) fudge going on here]

 Going back to Grice's formal systems and considering the question how to introduce an axiom of infinity, there are therefore a number of choices to be made.

The first is whether we just want there to be infinitely many things so that we could do arithmetic, or whether we want there to be things with infinite extensions (so we don't get confounded with finitists, or perhaps because we think that ordinary language is just as expressive as set theory).
To do the former, we could just add in the peano principles, to do the latter the language of sets and the assertion of a set closed under some successor relationship will do.

That was really the easy question.
Grice's system raised more difficult questions because it has different notions of existence and so when we assert the existence of sets we might or might not be doing something similar to asserting that Pegasus flies and hence, Grice insists, that somehow Pegasus exists.

So here is a question about Grice for Speranza.

When it comes to set theory in the context of his treatment of vacuous names, would Grice be thinking of sets as "logical fictions", and hence want to treat sets as non-denoting, or do we want sets to be more solid than that.
One issue which arises in this context is whether one can pile fictions on top of each other in the way that sets can be made from sets.
Can on have a fiction which is build out of fictions in some way?
Is it consistent with Grice's system that not only do sets fail to denote, but that their members lack denotations as well?

I'm afraid I have lost my grip on the system since it is years since I did my rehash (see: http://rbjones.com/rbjpub/pp/doc/t037.pdf) so I would have to spend some time getting it back into my head before I could see what the options are.

Meanwhile Speranza has launched us into extensionalism and other minimalisms (though perhaps he doesn't have more than one) which are an important area in which one might imagine Grice to have serious issue with Carnap, but in which I persist in thinking that Grice's does not see all the possibilities and that not all notions of extensionalism and not all practice of minimalism falls prey to his critiques, or ought by him to be considered objectionable.
So I am more inclined to chase this hare than get deeper into how infinity might work for Grice, which probably does get very complicated.

RBJ

Griceian infinity and the null set -- Ø --

Speranza

Further to previous commentary.

Part of the charm of "Vacuous Names" is how it connects with some demons within Grice -- and perhaps Carnaap.

In "Reply to Richards", Grice speaks of the demon (or bête noire) of Extensionalism, which may be made to connecd vis–à–vis certain points by Jones re: the policy of having the null set do duty for more things that it should!

On p. 68, Grice refers to  "Extensionalism" as a

"position imbued with teh spirit of Nominalism [another demon], and dear both to those who feel that 'Because it is red' is no more informative as an anser to the question 'Why is an English mail-box called 'red'?' than would 'Because he is Paul Grice' is an answer to the question 'Why is that distinguished-looking philosopher called 'Paul Grice'?', AND also to those whe are particularly impressed by the power of set theory."

----

Grice goes on

"The picture which, I suspect, is liable to go along with Extensionalism is that of the world of
PARTICULARS as a DOMAIN stocked with innumerable tiny pellets, internally INdistinguishable
from one another, BUT distinguished by the groups within which they fall, by the 'clubs' to which they belong."

"And since the clubs are distinguished ONLY by their memberships, there can

ONLY BE ONE CLUB TO WHICH NOTHING BELONGS."




The null set --  ø --


----


Grice goes on:

"As one might have predicted from the outset, this leads to the trouble when it comes to the
accomodation of EXPLANATION within such a system."



"Explanation of the ACTUAL presence of a particular feature"
in a particular subject depends
CRUCIALLY on the possibility of
saying what WOULD be the
consequence of the presence of such
and such features in that subject,
regardless of whether the
features in question even DO appear
in that subject, or indeed in ANY subject."



"On the face of it, if one adopts an
extensionalist viewpoint, the
presence of a feature in some particular
will have to be RE-EXPRESSED in terms
of that particular's membership of a
certain set; but if we proceed along
those lines, there
THERE IS ONLY ONE EMPTY SET,
the potential consequences of the
possession of in fact UNexemplified
features would be
INVARIABLY THE SAME,
now matter how different in meaning
the expressions used to specify
such features would ordinarily be
judged to be."

:

"I can think of TWO ways in which
to avoid [this unacceptable conclusion
of extensionalism -- as per above]."

Both ways, Grice says, "seem to me to suffer from serious drawbacks"

He cares them to expound them in some detail-

FIRST EXTENSIONALIST MANOUVRE with the null set:

"The first shows some degree of analogy

with a move which, as a matter of history,

was made by empiricists in connection with simple and complex ideas.

In that region an idea would be redeemed

from a charge of failure to conform to

empiricist principles though not being

derived from experience of its instantiating

particulars (there being no such particulars)

if it could be exhibited as a complex

idea whose component simple ideas were so derived."



Grice goes on:

"Somewhat similarly, the first proposal

seeks to

RELIEVE CERTAIN VACUOUS PREDICATES

or general terms from the

embarrasing consequence of

DENOTING the empty set

by exploiting the

NON-VACUOUSNESS of OTHER predicates

or general terms which are constitutents

IN THE DEFINITION of the original vacuous terms."



Grice goes on:

"(alpha) Start with TWO vacuous predicates,

say (ALPHA-1) 'is married to a daughter

of an English queen and a pope' and

(ALPHA-2) 'is a climber on hands and knees of

29,000 foot mountain."

"(BETA) If alpha-1 and alpha-2 are vacuous,

then the following predicates are

satisfied by the empty set Ø"




---

Grice continues:

(BETA-1) 'is a set composed of

daughters of an English queen and pope', and

(BETA-2) 'is a set composed of climbers

on hands and knees of a 29,000 foot moutantain".

Grice proceeds with a third step:

"(GAMMA) Provided R1 and R2 are suitably

interpreted, the predicates

beta-1 and beta-2 may be trated as

CO-EXTENSIVE

respectively with the following

REVISED predicates

'gamma-1' 'stands in R1 to a sequence

composed of the sets 'married to', 'daughters',

'English queens' and 'popes''

and

'gamma-2, 'stands in relation R2 to a

sequence composed of the set 'climbers',

'29,000 foot mountains', and 'things

done on hands and knees'."


The fourth step he calls delta.


"DELTA. We may FINALLY correlate with

the two initial predicates alpha-1 and alpha-2,

respectively, the following sequences

derived from gamma-1 and gamma-2:

delta-1, the sequence composed of the

relation R1 (taken in EXTENSION), the

set 'married to', the set 'daughters', the

set 'English queens', and the set 'popes';

and

delta-2: the sequence composed of the

relation R2, the set

'climbers', the set '29,000 foot mountains',

and the set 'things done on hands and knees'".


Grice goes on:


"These sequences are clearly distinct, and the proposal

is that THEY, rather than the EMPTY SET, should be

used for determining, in some way yet to be

specified, the explanatory potentialities of the

vacuous predicates alpha-1 and alpha-2."


Grice writes against this proposal: "My chief complaint against this proposal is that

it involves YET another commission of what I regard as one

of the main MINIMALIST sins, that of imposing

IN ADVANCE a limitation on the character of

explanations."

---

Why? Well, 'for it implicitly recognises it

as a condition on the propriety of using

vacuous predicates in explanation"

--- should that be proved!

"that the terms in question should be

representable as being correlated with

a sequence of NON-EMPTY sets."



Grice notes: "This is a condition which, I suspect, might

not be met by every vacuous predicate. But

the possibility of representing an explanatory

term as being, in this way or that,

reducible to some favoured iterm or types of

items should be a BONUS which some theories achieve,

demonstrating their elegance, not a condition of

eligibility for a particular class of would-be

explanatory items."


What about a second proposal then?

"The second suggested way of avoiding the

unwanted consequence is perhaps more intuitive

thatn the first. It certainly seems simpler."


Grice goes on: "The admissibility of vacuous predicates

in explanations of possible but non-actual

phenomena (why they would happen if they did happen),

depends, it is suggested, on the availability

of acceptable non-trivial generalisations wherein

which the predicate in question specifies the

antecedent condition."

Clear enough.

Grice goes on: "And, we may add, a generalisation whose

acceptability would be unaffected by any

variation on the specification of its antecedent

condition, provided the substitute were

vacuous, would certainly be trivial."



Grice goes on: "Non-trivial generalisations of this sort

are certainly available, if (1) they are

derivable as special cases from other generalisations

involving less specific antecedent conditions, and

(2) these other generalisations are adequately

supported by further specifics shose antecedent

conditions are expressed by means of non-vacuous

predicates."



Grice concludes about this second proposal: "The explanatory opportunities for vacuous

predicates depend on their embodiment in

a SYSTEM".

However, there's the drawback.

Grice notes: "My doubt about this second suggestion relate

to the STEPS which woud be needed in order

to secure an adequately powerful system."



Grice goes on:

"I conjecture, but cannot demonstrate, that

the ONLY WAY to secure such a system

would be to confer SPECIAL ONTOLOGICAL

privilege upon the entities of physical science"


Grice goes on:

"together with the system which that

science provides. But now a problem arises:

the preferred entities seem NOT to

be observable"





Grice goes on:

"or, in so far as they ARE observable,

their observability seesm to be more a

matter of conventional decisions

to COUNT such and such occurrences"



Grice goes on: "AS observations than it is a matter of

FACT. It looks as if states of affairs

in the preferred scientific world NEED,

for credibility, support from the

vulgar world of ordinary observation reported in the language of common sense."

-

Grice goes on:

"But to give THAT support, the

judgements and the linguistic usage

of the VULGAR nneds to be endowed

with a certain authority , which

as a matter of history"

--- things CAN change; people learn.

"the kind of minimalists whom I know or

know of have NOT seemed anxious

to confer."



To conclude: "But even if there WERE anxious

to confer it, what would validate

the conferring, since ex hypothesi it is NOT

the vulgar world but the specialist

scientific world which enjoys

ontological privilege? (If this objection

is sound, the second suggestion, like the

first, takes something which when present

is an assert, bouns, or embellishment, namely

systematicity, and under philosophical pressure

converts it into a necessity)."

And so on.

Griceian Infinity and the Empty Set -- ø --

Speranza

Just for the record then I am pasting R. B. Jones's two commentaries under different posts, as they deal with the connection of the idea of 'infinity' with that of 'empty set'.

The first commentary by R. B. Jones concerns this:

The axiom of infinity re

(AI)

"There is a set I ( which is postulated to be infinite), such that
-- the empty set is in I
 & such that
-- whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I.

Jones comments:

"Note that this formulation ... presumes that we have a constant whose name is the usual symbol for the empty set"

Viz.

 ø

"It might appear that this enables the existence of the empty set to be proven, but this is an illusion."

"Why?"

"Well, first of all, if our language L contains a constant C [and not just specific "ø"]  then we can prove "there exists x such that x = C" in first order logic, without benefit of any set-theoretic axioms at all."

I guess this relates to Grice's Vacuous Names, although his example there is "Marmaduke Bloggs", rather than the empty set -- although I think I have discussed with R. B. Jones the few references to the empty set by Grice in his "Reply to Richards" (which I should recheck) -- in Grice's criticism to extensionalism.

Jones goes on:

"However, we can't prove anything about C without some axioms, so we can't prove that C has no members. Mentioning C in the pivotal role in [AI] above makes no difference. You still can't prove   ø has no members from [AI]. In fact, [AI] works perfectly well whatever set plays that pivotal role."

"[AI] can be simplified so that it only states the existence of a NON-empty set closed under succession: the function from x to x u {x}."

"[AI], as stated, does not prove the existence of an empty set in a context in which that is not already provable. Well, not in a NON-CONTRIVED context. It would, of course, in a context in which there was an axiom (or theorem) asserting that the existence of I (the infinite set) entailed the existence of the empty set."

which then connects with the other post I had submitted. This referred to there existing an inductive set, whose members are

(i) the empty set;

(ii) for every member y of x, is also a member of x.

Infinity can be formulated so as to imply the existence of the empty set."

On that remark, Jones had commented:

"I find this observation rather puzzling ... since the existence of the empty set is usually taken to be an axiom ... even though it can be derived from other axioms. ...  The existence of

 ø

follows from the Separation Axiom Schema, AND from the Replacement Scheme (from which Separation can be obtained)."

"To understand what is intended here ... one really needs to know in WHAT context infinity [the set I] SUFFICES to derive the existence of [ø], since in the context of ZFC or NBG - {empty set, infinity} the existence of the empty set is already provable."

Which are excellent points and which I should be able to connect with Grice's 'infinitely many stars' -- or not! while I try to retrieve what Grice said on the infamous (is it?) null set.

Grice and Witters on 'infinitely many'

Speranza

Some [--people] like Witters, but Moore's MY man" (Austin, to Grice).

---

From:

Rodych, Victor, "Wittgenstein's Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Summer 2011 Edition), Edward N. Zalta (ed.), URL = .

http://plato.stanford.edu/entries/wittgenstein-mathematics/

"As in his intermediate position, the later [Witters] claims that

‘ℵ₀’

and “infinite series” get their mathematical uses from the use of ‘infinity’ in ordinary language (RFM II, §60).

-- where by ordinary language (or lingo) we mean things like German, or English.

"Although, in ordinary language, we often use ‘infinite’ and “infinitely many” as answers to the question “how many?,” and though we associate infinity with the enormously large, the principal use we make of ‘infinite’ and ‘infinity’ is to speak of the unlimited (RFM V, §14) and unlimited techniques (RFM II, §45; PI §218).

PUPIL: How many stars are there?

GRICE: There are infinitely many stars.

PUPIL: Infinitely many?

GRICE: As far as I know, yes.

"This fact is brought out by the fact “that the technique of learning ℵ₀ numerals is different from the technique of learning 100,000 numerals” (LFM 31).

"When we say, e.g.,

There are an infinite number of even numbers

or

There are infinitely many stars

we MEAN, as it were, that we have a mathematical technique or rule for generating even numbers [or stars?] which is limitless, which is markedly different from a limited technique or rule for generating a finite number of numbers, such as 1–100,000,000.

“We learn an endless technique,” says Wittgenstein (RFM V, §19), “but what is in question here is not some gigantic extension.”

"An infinite sequence, for example, is not a gigantic extension because it is not an extension, and ‘ℵ₀’ is not a cardinal number, for “how is this picture connected with the calculus,” given that “its connexion is not that of the picture | | | | with 4” (i.e., given that ‘ℵ₀’ is not connected to a (finite) extension)?"

 

"This shows, says Wittgenstein (RFM II, §58), that we ought to AVOID the word ‘infinite’ in mathematics wherever it seems to give a meaning to the calculus, rather than acquiring its meaning from the calculus and its use in the calculus."

 

IMPORTANT point above. Grice's discussion is other. He is concerned with the claim by Malcolm that ordinary language is sacred and free from self-contradictoriness. His example of the 'infinitely many stars' is one among many (four or five).

"Once we see that the calculus contains nothing infinite, we should not be ‘disappointed’ (RFM II, §60), but simply note (RFM II, §59) that it is not “really necessary… to conjure up the picture of the infinite (of the enormously big).”"

"A second strong indication that the later Wittgenstein maintains his finitism is his continued and consistent treatment of ‘propositions’ of the type “There are three consecutive 7s in the decimal expansion of π” (hereafter ‘PIC’).^"

 

^"n the middle period, PIC (and its putative negation, ¬PIC, namely, “It is not the case that there are three consecutive 7s in the decimal expansion of π”) is not a meaningful mathematical “statement at all” (WVC 81–82: Footnote #1)."

 

"On Wittgenstein's intermediate view, PIC—like FLT, GC, and the Fundamental Theorem of Algebra—is not a mathematical proposition because we do not have in hand an applicable decision procedure by which we can decide it in a particular calculus. For this reason, we can only meaningfully state finitistic propositions regarding the expansion of π, such as “There exist three consecutive 7s in the first 10,000 places of the expansion of π” (WVC 71; 81–82, Footnote #1)."

 

"The later Wittgenstein maintains this position in various passages in RFM (Bernays 1959 [1986, 176]). For example, to someone who says that since “the rule of expansion determine[s] the series completely,” “it must implicitly determine all questions about the structure of the series,” Wittgenstein replies: “Here you are thinking of finite series” (RFM V, §11). If PIC were a mathematical question (or problem)—if it were finitistically restricted—it would be algorithmically decidable, which it is not [(RFM V, §21), (LFM 31–32, 111, 170), (WVC 102–03)]."

 

"As Wittgenstein says at (RFM V, §9): “The question… changes its status, when it becomes decidable,” “[f]or a connexion is made then, which formerly was not there.”  And if, moreover, one invokes the Law of the Excluded Middle to establish that PIC is a mathematical proposition—i.e., by saying that one of these “two pictures… must correspond to the fact” (RFM V, §10)—one simply begs the question (RFM V, §12), for if we have doubts about the mathematical status of PIC, we will not be swayed by a person who asserts “PIC ∨ ¬PIC” (RFM VII, §41; V, §13)."

"Wittgenstein's finitism, constructivism, and conception of mathematical decidability are interestingly connected at (RFM VII, §41, par. 2–5)."

Witters:

"What harm is done e.g. by saying that God knows all irrational numbers?  Or: that they are already there, even though we only know certain of them?  Why are these pictures not harmless?"

 

For one thing, they hide certain problems.— (MS 124, p. 139; March 16, 1944)

"Suppose that people go on and on calculating the expansion of π."

 

"So God, who knows everything, knows whether they will have reached ‘777’ by the end of the world."

 

"But can his omniscience decide whether they would have reached it after the end of the world?"

 

"It cannot. I want to say: Even God can determine something mathematical only by mathematics.  Even for him the mere rule of expansion cannot decide anything that it does not decide for us."

"We might put it like this: if the rule for the expansion has been given us, a calculation can tell us that there is a ‘2’ at the fifth place."

 

"Could God have known this, without the calculation, purely from the rule of expansion?  I want to say: No. (MS 124, pp. 175–176; March 23–24, 1944)."

 

"What Wittgenstein means here is that God's omniscience might, by calculation, find that ‘777’ occurs at the interval [n,n+2], but, on the other hand, God might go on calculating forever without ‘777’ ever turning up."

"Since π is not a completed infinite extension that can be completely surveyed by an omniscient being (i.e., it is not a fact that can be known by an omniscient mind), even God has only the rule, and so God's omniscience is no advantage in this case [(LFM 103–04); cf. (Weyl, 1921 [1998, 97])]. "

"Like us, with our modest minds, an omniscient mind (i.e., God) can only calculate the expansion of π to some n^th decimal place—where our n is minute and God's n is (relatively) enormous—and at no n^th decimal place could any mind rightly conclude that because ‘777’ has not turned up, it, therefore, will never turn up."

Or not, i.e., or it will.

Grice -- the axiom of infinity and the von Neumann-Bernays-Gödel axioms -- infinity formulated so as to imply the existence of the empty set (cfr. Grice on "Vacuous Names")

Speranza

The axiom of infinity is also one of the von Neumann–Bernays–Gödel axioms.

In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of the canonical axiomatic set theory ZFC.

A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC.

The ONTOLOGY (as Grice would call it) of NBG includes:

proper classes,

objects having members but that cannot be members of other entities.

NBG's principle of class comprehension is predicative.

Quantified variables in the defining formula can range only over sets.

Allowing impredicative comprehension turns NBG into Morse-Kelley set theory (MK). NBG, unlike ZFC and MK, can be finitely axiomatized.

There exists an inductive set, namely a set x whose members are

(i) the empty set;

(ii) for every member y of x, is also a member of x.

Infinity can be formulated so as to imply the existence of the empty set.

REFERENCES

 

• Adámek, Jiří; Herrlich, Horst, and Strecker, George E (2004) [1990]. Abstract and Concrete Categories (The Joy of Cats) (PDF). New York: Wiley & Sons. ISBN 0-471-60922-6. 

• Bernays, Paul (1991). Axiomatic Set Theory. Dover Publications. ISBN 0-486-66637-9. 

• Ferreirós, José (2007), Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought (2nd revised ed.), Basel, Switzerland: Birkhäuser, ISBN 3-7643-8349-6 .

• Gödel, Kurt (1940), The Consistency of the Continuum Hypothesis, Princeton University Press .

• Hallett, Michael (1984), Cantorian Set Theory and Limitation of Size, Oxford: Clarendon Press .

• Kanamori, Akihiro (2009), "Bernays and Set Theory", Bulletin of Symbolic Logic 15: 43–69 . (Page numbering in Notes refers to online article whose numbering starts at 1.)

• Kanamori, Akihiro (2012), "In Praise of Replacement", Bulletin of Symbolic Logic 18: 46–90 .

• Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, North-Holland, ISBN 0-444-85401-0 .

• Mendelson, Elliott, (1997), An Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall. ISBN 0-412-80830-7. Pp. 225–86 contain the classic textbook treatment of NBG, showing how it does what we expect of set theory, by grounding relations, order theory, ordinal numbers, transfinite numbers, etc.

• Mirimanoff, Dmitry (1917), "Les antinomies de Russell et de Burali-Forti et le probleme fondamental de la theorie des ensembles", L'Enseignement Mathématique 19: 37–52 .

• Richard Montague, (1961), "Semantic Closure and Non-Finite Axiomatizability I," in Infinitistic Methods: Proceedings of the Symposium on Foundations of Mathematics, (Warsaw, 2–9 September 1959). Pergamon: 45-69.

• Muller, F. A., (2001), "Sets, classes, and categories," British Journal of the Philosophy of Science 52: 539-73.

• Müller, Gurt, ed. (1976), Sets and Classes: On the Work of Paul Bernays, Amsterdam: North Holland .

• Potter, Michael, (2004), Set Theory and Its Philosophy. Oxford Univ. Press.

• Pudlak, P., (1998), "The lengths of proofs" in Buss, S., ed., Handbook of Proof Theory. North-Holland: 547-637.

• von Neumann, John (1923), "Zur Einführung der transfiniten Zahlen", Acta litt. Acad. Sc. Szeged X. 1: 199–208 . English translation: van Heijenoort, Jean (1967), "On the introduction of transfinite numbers", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 346–354 .

• von Neumann, John (1925), "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik 154: 219–240 . English translation: van Heijenoort, Jean (1967), "An axiomatization of set theory", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 393–413 .

• von Neumann, John (1928), "Die Axiomatisierung der Mengenlehre", Mathematische Zeitschrift 27: 669–752 .

• von Neumann, John (1929), "Über eine Widerspruchsfreiheitsfrage in der axiomatischen Mengenlehre", Journal für die Reine und Angewandte Mathematik 160: 227–241 .

Grice and the axiom of infinity: "As far as I know, there are infinitely many numbers" (and "stars").

Speranza

In the formal language of the Zermelo–Fraenkel axioms, the axiom of infinity reads:

 there is a set I (the set which is postulated to be infinite), such that the empty set is in I and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I. Such a set is sometimes called an inductive set.

Robinson and Grice on the infinite

Speranza

Abraham Robinsohn was courted by Yale. Grice wasn't.

Abraham Robinson
Born	(1918-10-06)October 6, 1918 Waldenburg (Wałbrzych), German Empire
Died	April 11, 1974(1974-04-11) (aged 55) New Haven, Connecticut
Fields	Mathematics
Institutions	University of California, Los Angeles, Yale University
Alma mater	Hebrew University, University of London
Doctoral advisor	Paul Dienes
Doctoral students	Azriel Levy, Peter Winkler, A. H. Lightstone
Known for	Non-standard analysis
Influences	Gottfried Leibniz, Abraham Fraenkel

Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) is a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics.

 

Robinsohn was born to a Jewish family with strong Zionist beliefs, in Waldenburg, Germany, which is now Wałbrzych, in Poland.

In 1933, he emigrated to British Mandate of Palestine, where he earned a first degree from the Hebrew University.

Robinsohn was in France when the Nazis invaded during World War II, and escaped by train and on foot, being alternately questioned by French soldiers suspicious of his German passport and asked by them to share his map, which was more detailed than theirs.

While in London, Robinsohn joined the Free French Air Force and contributed to the war effort by teaching himself aerodynamics and becoming an expert on the airfoils used in the wings of fighter planes.

After the war, Robinsohn worked in London, Toronto, and Jerusalem, but ended up at University of California, Los Angeles in 1962.

 

Robinsohn become known for his approach of using the methods of mathematical logic to attack problems in analysis and abstract algebra.

Robinsohn introduced many of the fundamental notions of model theory.

 Using these methods, Robinsohn finds a way of using formal logic to show that there are self-consistent nonstandard models of the real number system which include infinite and infinitesimal numbers.

Others, such as Wilhelmus Luxemburg, showed that the same results could be achieved using ultrafilters, which made Robinsohn's work more accessible to mathematicians who lacked training in formal logic.

Robinsohn's book Non-standard Analysis was published in 1966.

Robinsohn was strongly interested in the history and philosophy of mathematics, and often remarked that he wanted to get inside the head of Leibniz, the first mathematician to attempt to articulate clearly the concept of infinitesimal numbers.

He meant it 'metaphorically', he later explained ("as if "per implicatura"").

While at UCLA Robinsohn's colleagues remember him as working hard to accommodate PhD students of all levels of ability by finding them projects of the appropriate difficulty.

Robinsohn was courted by Yale, and after some initial reluctance, he moved there in 1967.

He died of pancreatic cancer in 1974.

Notes

1. ^ Hodges, W: "A Shorter Model Theory", page 182. CUP, 1997

Publications

 

Robinson, Abraham (1963), Introduction to model theory and to the metamathematics of algebra, Amsterdam: North-Holland, ISBN 978-0-7204-2222-1, MR 0153570 

Robinson, Abraham (1977) [1956], Keisler, H. Jerome, ed., Complete theories, Studies in Logic and the Foundations of Mathematics (2nd ed.), Amsterdam: North-Holland, ISBN 978-0-7204-0690-0, MR 0472504 

Robinson, Abraham (1979), Keisler, H. Jerome, ed., Selected papers of Abraham Robinson. Vol. I Model theory and algebra, Yale University Press, ISBN 978-0-300-02071-7, MR 533887 

Robinson, Abraham (1979), Luxemburg, W. A. J.; Körner, S., eds., Selected papers of Abraham Robinson. Vol. II Nonstandard analysis and philosophy, Yale University Press, ISBN 978-0-300-02072-4, MR 533888 

Robinson, Abraham (1979), Young, A. D., ed., Selected papers of Abraham Robinson. Vol. III Aeronautics, Yale University Press, ISBN 978-0-300-02073-1, MR 533889 

Robinson, Abraham (1996) [1966], Non-standard analysis, Princeton Landmarks in Mathematics (2nd ed.), Princeton University Press, ISBN 978-0-691-04490-3, MR 0205854 

See also[edit]

• Influence of non-standard analysis

References[edit]

• J. W. Dauben, Abraham Robinson: The Creation of Nonstandard Analysis, A Personal and Mathematical Odyssey, Princeton, NJ: Princeton University Press, 1998

External links[edit]

• O'Connor, John J.; Robertson, Edmund F., "Abraham Robinson", MacTutor History of Mathematics archive, University of St Andrews .
• Abraham Robinson at the Mathematics Genealogy Project
• Abraham Robinson — Biographical Memoirs of the National Academy of Sciences

[hide] v t e Infinitesimals History Adequality Infinitesimal calculus Leibniz's notation Integral symbol Criticism of non-standard analysis The Analyst The Method of Mechanical Theorems Cavalieri's principle Related branches Non-standard analysis Non-standard calculus Internal set theory Synthetic differential geometry Constructive non-standard analysis Formalizations Differentials Hyperreal numbers Dual numbers Surreal numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of Continuity Overspill Microcontinuity Transcendental law of homogeneity Scientists Gottfried Wilhelm Leibniz Abraham Robinson Pierre de Fermat Augustin-Louis Cauchy Leonhard Euler Physics Engineering Infinitesimal strain theory Infinitesimal oscillations of a pendulum Textbooks Analyse des Infiniment Petits Elementary Calculus

Authority control VIAF: 59260725

Persondata
Name	Robinson, Abraham
Alternative names
Short description	American mathematician
Date of birth	October 6, 1918
Place of birth	Waldenburg (Wałbrzych), Germany
Date of death	April 11, 1974
Place of death	New Haven, Connecticut

      

Categories:

• Mathematics of infinitesimals
• 1918 births
• 1974 deaths
• 20th-century mathematicians
• Alumni of the University of London
• American mathematicians
• German mathematicians
• German Jews
• German emigrants to the United States
• American people of German-Jewish descent
• People from Wałbrzych
• University of California, Los Angeles faculty
• Yale University faculty
• Mathematical logicians
• Model theorists
• Institute for Advanced Study visiting scholars