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Harold R. (Hal)
Foster’s Prince Valiant
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(5) The number of syllables in the English names of
finite integers tends to increase as the integers grow larger, and must
gradually increase indefinitely, since only a finite number of names can be
made with a given finite number of syllables. Hence the names of some integers
must consist of at least nineteen syllables, and among these there must be a least. Hence "the least integer not nameable in fewer than nineteen syllables"
must denote a definite integer; in fact, it denotes 111,777. But "the
least integer not nameable in fewer than nineteen syllables" is itself a name consisting of eighteen syllables;
hence the least integer not nameable in fewer than nineteen syllables can be
named in eighteen syllables, which is a contradiction*1.
<<(5) The paradox about "the least integer
not nameable in fewer than nineteen syllables" embodies, as is at once
obvious, a vicious-circle fallacy. For the word "nameable" refers to the totality of names, and yet is allowed
to occur in what professes to be one among names. Hence there can be no such
thing as a totality of names, in the sense in which the paradox speaks of
"names." It is easy to see that, in virtue of the hierarchy of
functions, the theory of types renders a totality of "names"
impossible. We may, in fact, distinguish names of different orders as follows: (a) Elementary names will be such as
are true "proper names," i.e.
conventional appellations not involving any description. (b) First-order names will be such as
involve a description by means of a first-order function; that is to say, if φ!x^ is a first-order function, "the
term which satisfies φ!x^" will
be a first-order name, though there will not
always be an object named by this name. (c) Second-order names will be such as involve a description by means
of a second-order function; among such names will be those involving a reference
to the totality of first-order names. And so we can proceed through a whole
hierarchy. But at no stage can we give a meaning to the word
"nameable" unless we specify the order of names to be employed; and
any name in which the phrase "nameable by names of order n" occurs is necessarily of a
higher order than the nth. Thus the
paradox disappears.
(6) Among trans-finite ordinals some can be defined,
while others cannot; for the total number of possible definitions is À0*2, while the number
of trans-finite ordinals exceeds À0.
Hence there must be indefinable ordinals, and among these there must be a
least. But this is defined as "the
least indefinable ordinal," which is a contradiction*3.
(7) Richard's paradox*4 is akin to that of the least
indefinable ordinal. It is as follows: Consider all decimals that can be
defined by means of a finite number of words; let E be the class of such decimals. Then E has À0
terms; hence its members can be ordered as the 1st, 2nd, 3rd, .... Let N be a number defined as follows: If the nth figure in the nth
decimal is p, let the nth figure in N be p + 1 (or 0, if p = 9). Then N is different from all the members of E, since, whatever finite value n
may have, the nth figure in N is different from the nth figure in the nth of the decimals composing E,
and therefore N is different from the
nth decimal. Nevertheless we have
defined N in a finite number of words, and therefore N ought to be a member of E.
Thus N both is and is not a member of
E.
<<(6,7) The solutions of the paradox about the
least indefinable ordinal and of Richard's paradox are closely analogous to the
above. The notion of "definable," which occurs in both, is nearly the
same as "nameable," which occurs in our fifth paradox:
"definable" is what "nameable" becomes when elementary
names are excluded, i.e.
"definable" means "nameable by a name which is not
elementary." But here there is the same ambiguity as to type as there was
before, and the same need for the addition of words which specify the type to
which the definition is to belong. And however the type may be specified,
" the least ordinal not definable by definitions of this type" is a
definition of a higher type; and in Richard's paradox, when we confine
ourselves, as we must, to decimals that have a definition of a given type, the
number N, which causes the paradox,
is found to have a definition which belongs to a higher type, and thus not to
come within the scope of our previous definitions.
In all the above contradictions (which are merely
selections from an indefinite number) there is a common characteristic, which
we may describe as self-reference or reflexiveness. The remark of Epimenides
must include itself in its own scope. If all
classes, provided they are not members of themselves, are members of w, this must also apply to w; and similarly for the analogous
relational contradiction. In the cases of names and definitions, the paradoxes
result from considering non-name-ability and indefinability as elements in
names and definitions. In the case of Burali-Forti's paradox, the series whose
ordinal number causes the difficulty is the series of all ordinal numbers. In
each contradiction something is said about all
cases of some kind, and from what is said a new case seems to be generated,
which both is and is not of the same kind as the cases of which all were
concerned in what was said. But this is the characteristic of illegitimate
totalities, as we defined them in stating the vicious-circle principle. Hence
all our contradictions are illustrations of vicious-circle fallacies. It only
remains to show, therefore, that the illegitimate totalities involved are
excluded by the hierarchy of types which we have constructed.
An indefinite number of other contradictions, of
similar nature to the above seven, can easily be manufactured. In all of them,
the solution is of the same kind. In all of them, the appearance of
contradiction is produced by the presence of some word which has systematic
ambiguity of type, such as truth, falsehood, function, property, class, relation, cardinal, ordinal, name, definition. Any
such word, if it’s typical ambiguity is overlooked, will apparently generate a
totality containing members defined in terms of itself, and will thus give rise
to vicious-circle fallacies. In most cases, the conclusions of arguments which
involve vicious-circle fallacies will not be self-contradictory, but wherever
we have an illegitimate totality, a little ingenuity will enable us to
construct a vicious-circle fallacy leading to a contradiction, which disappears
as soon as the typically ambiguous
words are rendered typically definite, i.e.
are determined as belonging to this or that type.
Thus the appearance of contradiction is always due to
the presence of words embodying a concealed typical ambiguity, and the solution
of the apparent contradiction lies in bringing the concealed ambiguity to
light.
In spite of the contradictions which result from
unnoticed typical ambiguity, it is not desirable to avoid words and symbols
which have typical ambiguity. Such words and symbols embrace practically all
the ideas with which mathematics and mathematical logic are concerned: the
systematic ambiguity is the result of a systematic analogy. That is to say, in
almost all the reasoning which constitutes mathematics and mathematical logic,
we are using ideas which may receive any one of an infinite number of different
typical determinations, any one of which leaves the reasoning valid. Thus by
employing typically ambiguous words and symbols, we are able to make one chain
of reasoning applicable to any one of an infinite number of different cases,
which would not be possible if we were to forego the use of typically ambiguous
words and symbols.
Among propositions wholly expressed in terms of
typically ambiguous notions practically the only ones which may differ, in
respect of truth or falsehood, according to the typical determination which
they receive, are existence-theorems. If we assume that the total number of
individuals is n, then the total
number of classes of individuals is 2n",
the total number of classes of classes of individuals is 22", and so on. Here n may be either finite or infinite, and in either case 2n >n. Thus cardinals
greater than n but not greater than 2n exist as applied to
classes, but not as applied to classes of individuals, so that whatever may be
supposed to be the number of individuals, there will be existence-theorems
which hold for higher types but not for lower types. Even here, however, so
long as the number of individuals is not asserted, but is merely assumed
hypothetically, we may replace the type of individuals by any other type,
provided we make a corresponding change in all the other types occurring in the
same context. That is, we may give the name "relative individuals"
to the members of an arbitrarily chosen type Ï„,
and the name "relative classes of individuals" to classes of
"relative individuals," and so on. Thus so long as only hypotheticals
are concerned, in which existence-theorems for one type are shown to be implied
by existence-theorems for another, only relative types are relevant even in
existence-theorems. This applies also to cases where the hypothesis (and
therefore the conclusion) is asserted, provided the assertion holds for any
type, however chosen. For example, any type has at least one member; hence any
type which consists of classes, of whatever order, has at least two members.
But the further pursuit of these topics must be left to the body of the work.
*1
This contradiction was suggested to us by Mr G. G. Berry of the Bodleian
Library.
*2 À0 is
the number of finite integers. *123.
*3
Cf. Konig, "Ueber die Grundlagen der Mengenlehre und das
Kontinuumproblem," Math. Annalen, Vol. LXI. (1905); A. C. Dixon, "On
'well-ordered' aggregates," Proc. London Math. Soc. Series 2, Vol. iv.
Part i. (1906); and E. W. Hobson, "On the Arithmetic Continuum,"
ibid. The solution offered in the last of these papers depends upon the variation
of the " apparatus of definition," and is thus in outline in
agreement with the solution adopted here. But it does not invalidate the
statement in the text, if " definition " is given a constant meaning.
*4
Cf. Poincare, "Les mathematiques et la logique," Revue de
Metaphysique et de Morale, Mai 1906, especially sections vii. and ix.; also
Peano, Revista de Mathematica, Vol. viii. No. 5 (1906), p. 149 ff.
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| Script: Stan Lee Pencils: & Inks: Steranko © Respective copyright/trademark holders. |
This part, addressing contradictions, is key to translating the use and
real world value of the system into practice. This is also the clearest explanation
thus far of the basic understanding of shaman sight; or seeing through
externalities that distract most human beings from using their ability to recognize complex and difficult
problems, let alone thinking them through and separating their causes and
solutions.
When we want to discern reality in contradictory intentions no matter how large or small, sorting it out is a matter of looking at all sides of the issue without prejudice; propositions placed in evidence that include arguments that are self-referential are the ones that unravel as vicious-circle Fallacies. Don’t presume these gremlins: truth, falsehood, function, name, reality, relation, etc. must be intentionally removed, though perhaps they may be, these typical ambiguities may foul the water but conversely they allow us to open the tap of creativity that makes this system and all mathematics happen in the first place.
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Script and Layouts: Jim Shooter Pencils
and Inks: George Papp Cosmic Boy Phantom
Girl & LSHTM ©
Respective copyright/trademark holders.
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Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders.
|
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