Monday, February 22, 2016

What do we mean by "finite"?

The standard mathematical definition is a set is finite provided that its cardinality is a natural number. But what are the natural numbers? Think about all the non-standard models of the natural numbers, models of arbitrarily high cardinality, and note that if we use one of these high-cardinality models as what we mean by "natural number", we will get a different extension for "finite". So how do we manage to pick out one particular meaning for "finite" or "natural number"? I want to offer four options.

1. Physics. The world gets described by a lot of equations. In these equations, mathematical objects (numbers, Hilbert spaces, etc.) represent features of the real world. We then constrain our interpretation of mathematical terms like "finite", "natural number" and "real number" by the requirement that the mathematical objects correspond as closely as possible to the features of the real world. In other words, we privilege those models of the natural numbers which fit the physical world. Note that this very much requires a significant dollop of scientific realism.

2. The future. We use the infinity of future days to define "finite": a natural number is any number that we will ever reach by counting once per day. This requires the actual world to have an infinite future.

3. Causal finitism. According to causal finitism, no object has an infinite causal history. But, very plausibly, any finite number of causes can be the causal history of an event. We can use these modal claims to constrain the interpretation of "finite".

4. God. Maybe God simply chose one extension for "finite", "natural number", etc., and made our words correspond to that. Cf. this.


Drew said...

Couldn't you also say that a set is finite if the cardinality of any of its proper subsets is less than the cardinality of the set?

Or maybe if no proper subset has the same cardinality as the set. I am not aware of any counterexamples to this latter definition.

Heath White said...

I think #2 is an instance of #1, where "future days" is the privileged physical thing being counted. As such, it seems somewhat arbitrary. For #3, it is probably going to be quite difficult to fix constraints on a possible causal history which are more constraining than possible models of natural numbers. I.e. why are we so sure that a causal history couldn't have a high-cardinality-finite number of links in its causal chain? And #4 strikes me as implausible--and not helpful for theorizing--in this case as in all the others.

It seems to me that we have a much better grasp on natural numbers than on any formal model of them. (Is it "What Numbers Could Not Be" that argues that, since there are multiple incompatible models of the natural numbers in set-theoretic terms, no set-theoretic model IS the natural numbers?) So what we mean by 'finite' is the thing we grasp when we learn the natural numbers. I suspect that is a version of your #1.

Alexander R Pruss said...

You could, but there are two problems.

The less serious one is that this requires some version of the Axiom of Choice. Without the Axiom of Choice, one can have infinite sets that are Dedekind-finite (i.e., larger than a proper subset).

The more serious is that Loewenheim-Skolem type of reasoning still applies. Given the Axiom of Choice, your definition is equivalent to "cardinality is a natural number". But assuming ZFC is consistent, one can reinterpret all of ZFC in such a way that "natural numbers" refers to a set of high cardinality.

(Quick proof: Let n be a high cardinality. Introduce n new symbols c_i into the language, where i ranges over a set I of size n. Add to the axioms all the instances of the statements: "c_i is a natural number" (for i in I) and "c_i is not equal to c_j" (for i and j distinct members of I). Every finite subset of the axioms is consistent. By the Compactness Theorem, there is still a model. But in this model the set corresponding to the natural numbers will have to contain at least n elements.)

And under such a reinterpretation, your definition will also give the same unhappy results as the natural number definition.

Alexander R Pruss said...


Maybe, but it's not clear that the future days are a part of physics. A plausible reading of Scripture suggests that there will be an end of this universe, and then there will be a new universe ("a new heavens and a new earth") for the saints. If so, then our universe's future days are finite, and it is not clear that the physics of our universe describes the new universe. "Our future days" then includes both a finite number of days in this universe AND an infinite number of days in the new universe, and so it goes beyond physics.

One might, of course, try a counterfactual rendering: maybe the natural numbers correspond to the infinity of future days that there *would be* if our universe were not to have an end.

"It seems to me that we have a much better grasp on natural numbers than on any formal model of them."

That may be, but it's very hard to see how our language manages to gain reference to *the* natural numbers. The theistic story could work. Maybe there is some magical theory of reference that works, too?

I do, think, though that there are some more options I should have listed:

5. The natural numbers are natural in the David Lewis sense of the word, and the exotic models are not natural.

6. Sethood and the set membership relation are natural in the David Lewis sense of the word.

But these options may require Platonism.

Drew said...

I had thought that, according to Hannes Leitgeb of LMU, an infinite set is literally defined as a set which has a proper subset of the same cardinality. What I don't understand is how any set with a cardinality equal to or greater than aleph-null could possibly lack a proper subset with the same cardinality.

Besides, couldn't we stipulate that finite means a cardinality less than aleph-null. Even if it's possible for "natural numbers" to be defined as a set of high cardinality, the "less than aleph-null" definition certainly cannot.

Alexander R Pruss said...

1. "A set which has a proper subset of the same cardinality" is a definition of a Dedekind-infinite set, not just an infinite set. If the Axiom of Choice is false, there might be infinite sets that aren't Dedekind-infinite. "What I don't understand is how any set with a cardinality equal to or greater than aleph-null could possibly lack a proper subset with the same cardinality." It couldn't. But if the Axiom of Choice is false, there might be an infinite set whose cardinality is neither equal to, nor greater than, nor less than aleph-null.

2. "Besides, couldn't we stipulate that finite means a cardinality less than aleph-null." Sure. But then we have the question of what "aleph-null" refers to. There are exotic models of set theory where it refers to something really big.

Drew said...
This comment has been removed by the author.
Drew said...

Ok, we can define finite as a set that, given the axiom of choice, whose proper subsets all have a lower cardinality than the set.

If one can interpret ZFC so that "natural numbers" refer to sets of high cardinality, then this definition cannot be equivalent to it, since it is independent of what we mean by "natural numbers"

Kolten Ellis said...

Regarding #2, this may be an argument agains the possibility of an actual infinite- if there were an actually infinite future, you could count an anctuakly infinite number of numbers. The number of numbers counted is numerically identical to the final number reached in the sequence (assuming that one counts in sequence). Thus, if the future were actually infinite, you could count to infinity. But if a finite number is defined as one that you could count to, counting one per day, then infinity becomes a finite number, which is absurd. The future could be potentially infinite, but not actually infinite.

Alexander R Pruss said...

Mr Ellis:

"The number of numbers counted is numerically identical to the final number reached in the sequence": This assumes that the sequence has a final number.

Kolten Ellis said...

I probably should have said something like, "the most recent number counted too." I think the argument still holds, although I suppose if we count to a natural number larger than the previous one, we haven't achieved the number whose set includes all natural numbers, and thus have not counted to infinity.

Richard Davis said...

If the term 'positive number' is more basic than the term 'positive finite number,' won't this work?

Aleph-null is the least positive number such that there are no two lesser positive numbers of which it is the sum.

Then a finite positive number is any positive number less than Aleph-null.

I think we can also revise the definition slightly in order to make do with the term 'positive whole number' instead of 'positive number':

Aleph-null is the least positive whole number n other than 1 such that there is no positive whole number m < n and positive whole number o < n such that n = m + o.

And 'positive whole number' can be defined thus:

x is a positive whole number iff there are some things each of which is qualitatively identical to 1 and x is their mereological sum.

What this scheme takes as basic is 1 and a theory of numbers as mereological sums of things qualitatively identical to 1.

Richard Davis said...

Sorry -- last paragraph should read:

What this scheme takes as basic is 1 and a theory of WHOLE numbers as mereological sums of things qualitatively identical to 1.

Alexander R Pruss said...

You still need a relation of "same size as" between your whole numbers, since if x, y and z are qualitatively identical to 1, then there may well be be several twos, such as x+y, x+z and y+z. They are all "the same size", of course.

You may be able to use second-order quantification to define that relation: two whole numbers are the same size iff there is a one-to-one and onto relation between the unit parts of the numbers (where the unit parts are the parts qualitatively identical to 1). But I think using second-order quantification is cheating, but that's a long story.

Richard Davis said...

I was thinking that in that sort of case, one pair would be the fundamental one and the others would be its echoes. The fundamental one will be the only one that is a number.

I think we can define which one is fundamental without using any primitives besides 1, 'mereological sum' and 'numerically distinct.' This is a theory on which whenever there are some entities which are qualitatively identical, there is some one entity (the 'root') which is qualitatively identical to each of them but not numerically distinct from any of them.


'x branches from y' := 'There is some z such that x and z are numerically distinct, but neither is numerically distinct from y'

'x is a root' := 'There is nothing from which x branches'

'x, y, z, ... are qualitatively identical' := 'There is some root which is not numerically distinct from any of x, y, z, ... "

'... is an 1-shadow' := '... is qualitatively identical to 1'

'... is a 1-concatenation' := '... is a mereological sum of some 1-shadows'

'... is a positive whole number' := '... is both a 1-concatenation and a root'

This sequence defines 'positive whole number' down into no non-logical primitives other than 1, mereological summation, and numerical distinctness.

It does mean the relationship of 'not being numerically distinct from each other' fails to be transitive. But I think that's fine, since numerical identity can easily be identified as the relationship 'being numerically distinct from all the same things' (which IS transitive) rather than the relationship 'not being numerically distinct from each other.'

Aristotle has something relevant to say about this (I think?) with regard to two numerically distinct routes (Athens to Sparta, Sparta to Athens), each of which 'is' (in some strong sense of is) the one road between the two cities.

Richard Davis said...

One edit:

I said we should define numerical identity as 'being numerically distinct from all the same things'. But there's an alternate view:

'not being numerically distinct' and 'being numerically identical' ARE the very same relationship, and so neither is transitive. But there is a transitive relationship (one that has to do with identity) in the vicinity, namely, the relationship 'being numerically identical to all the same things.'

Call this 'super identity.' It's not quite the same relationship as 'not being numerically distinct.' On the theory I'm giving, super identity is transitive: given how I've defined it, the fact that it is transitive is a theorem of first order logic. By contrast, on this theory, numerical identity is not transitive, and in at least one respect this is a good thing: The transitivity of numerical identity is neither a theorem of more basic principles of logic nor explicable in any other sort of way that I can easily see. So if we had to accept that numerical identity was transitive, we'd be having to accept a fact for which there was no explanation. Instead, on the present theory, we get to ditch this fact and take a theorem of logic (the transitivity of super identity) in its place. That's a win.

The trick, then, is to say that our intuitions to the effect that numerical identity is transitive arise largely either from (i) cognitively confusing super identity with numerical identity or (ii) systematically failing to consider the rare cases (roots and shadows, roads and routes) where to the transitivity of numerical identity there are plausible counterexamples.

Also nice to have a real definition of 'qualitative identity' instead of taking it as some sort of primitive, or defining it in terms of the primitive term 'quality' (this theory doesn't need 'quality' as a primitive term: it uses 'root' instead, and that term is fully defined down into other more basic terms rather than being a primitive).