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I would prefer to ask this in the math community, but that crowd is hostile toward anything hinting of philosophy. It is my contention that a construction of the real number system which begins with the most primitive concepts will begin by constructing the natural numbers beginning with the number one.

Can zero be defined without some definition of one? Can one be defined without some definition of zero?

This is an "after the fact" edit; since I've already accepted an answer. I just want to add this to save other's the trouble of bringing to my attention the Dedekind-Peano method of constructing the natural or whole numbers.

These are my current notes on the construction of the real numbers using Peano's axioms with 1 as the non-successor:

https://drive.google.com/open?id=1HRn2OJJV8OiJNBvDBTpsSjDL0VHdLvbE

These are my current notes on constructing the real numbers using a Peano-like set of axioms with 0 as the non-successor:

https://drive.google.com/open?id=1gB7yd8acGcxA1a9Gekz5GGf7HY87U1Sw

The notes are guaranteed to contain errors, dubious observations, idiosyncratic notation and methods, significant redundancy, fuzzy logic, etc.

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    In standard ZF (Zermelo-Fraenkel) set theory, the number 0 is defined as the empty set. Then 1 is defined as the set containing 0. 2 is defined as the set containing 0 and 1, and so forth. So yes in fact 0 is the first thing that gets defined in the absence of everything else. The big trick is the axiom of infinity, which lets us put 0, 1, 2, 3, ... all together into a set. Once we do that we have a model of the Peano axioms and we're off to the races.
    – user4894
    Commented May 22, 2019 at 23:41
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    @Conifold These are my current notes on the construction of the real numbers using Peano's axioms with 1 as the non-successor: drive.google.com/open?id=1HRn2OJJV8OiJNBvDBTpsSjDL0VHdLvbE These are my current notes on constructing the real numbers using a Peano-like set of axioms with 0 as the non-successor: drive.google.com/open?id=1gB7yd8acGcxA1a9Gekz5GGf7HY87U1Sw
    – Steven Thomas Hatton
    Commented May 23, 2019 at 3:23
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    @Conifold My point is that we are already employing a vast number of concepts when we undertake the formalization of mathematics. Whether there is a part of speech called an article in a given language really isn't that significant. There will be some concept of unitary existent entities. These are called "things" in English. Before we can consider a question such as "how many things are are in this room?" we have to have a concept of a "thing". And things are naively countable. I suspect that the etymology of word "none" will show it to be derived from the word "one". And "nothing"?
    – Steven Thomas Hatton
    Commented May 23, 2019 at 4:03
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    You are mixing meta-use of a concept with its linguistic use. Do we need some extra concepts to describe a formal language? Sure. But do we need them to set it up? No. What we really need is a practical skill for manipulating the symbols properly. That verbalizing this skill requires "things", or something else, is moot. It is well-known that historically humans had hard time with the concept of zero. Does it tell us something about our cognitive and linguistic apparatus? Definitely. Does etymology/history/learning habits tell us anything about the logical order of concepts? No.
    – Conifold
    Commented May 23, 2019 at 4:34
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    What we find conceivable (even as a species) is only a testament to (the lack of) our imagination, which historically proved to be quite poor. The the concept of tabula rasa presupposes nothing of the sort, only the etymology of its label does. As for Cartesian self-evidence and Kantian a priori, few take them seriously today. Non-Euclidean geometries, Frege's inconsistent "basic laws of thought", and incompleteness, to name a few, exposed the aforementioned lack of imagination very vividly. You really should read Azzouni before continuing.
    – Conifold
    Commented May 23, 2019 at 4:59

2 Answers 2

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According to the empirical research of the Natural Semantic Metalanguage project, there are a group of around 65 "semantic primes". These are core concepts shared by all human languages, which are the basis of all other meanings, and which cannot be usefully broken down into other concepts; any definitions of these primes will inevitably end up more convoluted than the word itself. (If this is ever not the case then that is taken as evidence that the roster of primes needs revising. While it changed rapidly in early years, it is now quite stable.)

Two of these primes are numbers: ONE and TWO. This indicates that from a linguistic perspective these two numbers are fundamental, and all languages will have lexical items for these primes. Zero is not a prime, indicating that it is derived and defined from these primes, and that languages may not have simple ways of referring to 0, which is indeed the case.

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    +1... because... I've never heard of semantic primes and they are bloody interesting...
    – Adam Sharpe
    Commented May 23, 2019 at 16:20
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This answer will consider ways that zero and one need to be described in terms of each other using the axioms of Peano arithmetic.

Wikipedia describes a model of the axioms of Peano arithmetic as a triple:

A model of the Peano axioms is a triple (N, 0, S), where N is a (necessarily infinite) set, 0 ∈ N and S : N → N satisfies the axioms above.

The first axiom, 0 is a natural number, appears to define 0 independently of 1, but S(0) = 1 shows that 1 is not described independently of 0.

However, given 0 and S(0)=1 as natural numbers, that is, elements of N, how do we know that 0 does not equal S(0)? Since elements of a set must be distinct, how do we know that there isn't only one element in N?

The equality relation is defined through four axioms insuring that it is reflexive, symmetric, transitive and that the natural numbers are closed under equality. The eighth axiom guarantees the distinctness of 0 and 1 given the axioms for the equality relation:

For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0.

So 0 does not equal 1, but this describes 0 in terms of the existence of S(0)=1.

So, if we have a set of more than one element containing zero and one these two elements will need to be described as distinct from each other and hence they will need to be described in terms of each other guaranteeing that the equality relation does not hold between them.

Wikipedia contributors. (2019, May 20). Peano axioms. In Wikipedia, The Free Encyclopedia. Retrieved 10:52, May 23, 2019, from https://en.wikipedia.org/w/index.php?title=Peano_axioms&oldid=897956091

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  • I'm sorry for not providing more background. I've added to my original question some links to my personal notes on applying the Dedekind-Peano method of constructing the natural or whole numbers. My question was intended to address things at a more fundamental level. There are, IMO, good structural arguments for favoring 1 as the non-successor. My effort to list my arguments in favor of 1 as the first number is what motivated my question. Though there is no "final word" in Philosophy, I really like the answer I accepted. One more comment to follow.
    – Steven Thomas Hatton
    Commented May 23, 2019 at 18:01
  • Notice that the discussion of Peano's axioms freely uses concepts involving numbers. One argument favoring 0 as the non-successor, given by Knuth, is that doing so is consistent with computer math. IMO, that is really sleight of hand. It takes at least one bit to represent 0. That is a variant of my contention that the natural numbers should be self-counting. Beginning with 1 will produce a self-consistent, useful number system which (I believe) gives the same results obtainable from a system with 0 as the non-successor. Except the additive identity mapping takes a different form.
    – Steven Thomas Hatton
    Commented May 23, 2019 at 18:33
  • @StevenThomasHatton I think the reason to start with 0 is to have the additive identity in the set. According to the Wikikpedia article I cited, Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number. You should be able to start with 1.
    – Frank Hubeny
    Commented May 23, 2019 at 18:56
  • You certainly can start with 1. I call the result the natural numbers. The system beginning with 0, I call the whole numbers. Once the natural numbers are constructed, a reasonable next step is to seek an extended domain of numbers so that equations $a=x+b$ can be solved for $x$, even when $a\le b$. The extended domain is the set of integers, which includes 0. But the integers are fully expressible as pairs of natural numbers. The integer zero is just the equivalence class of all pairs $\left\langle x,x\right\rangle $. Which is a fancy way of saying $x-x=0$.
    – Steven Thomas Hatton
    Commented May 23, 2019 at 19:09

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