One obvious feature is the central importance of lack of an upper bound in true number competence (in something like the way that our sentence understanding and sentence generating competence does not intrinsically have a maximum length bound, though various practical problems may impose length bounds. (This is essentially a generalisation of Chomsky's old distinction between grammatical competence (potentially infinite) and grammatical performance (bounded by short term memory, brain size, etc.). The first competence could be possesed by a child with a grasp of an unbounded iterative or recursive mechanism for generating number names. Without various competences related to the uses of those names, the child would not have number competences. I gave an introduction to some of those competences and ways in which they could be implemented in a computer capable of generating parallel synchronised systematic actions in chapter 8 of my 1978 book http://www.cs.bham.ac.uk/research/projects/cogaff/crp/#chap8 Unfortunately, in 1978 I did not make clear enough that I was assuming another requirement, known to David Hume, Gottlob Frege, Bertrand Russell, Jean Piaget and many others, but I later found was ignored by many researchers into number competences: namely an understanding that one-one correspondence (bijection) is a *necessarily* transitive relation. (It is also symmetric and reflexive, but those are simpler conditions.) Frege and Russell/Whitehead tried to show that using pure logic we can define a concept of number that includes 1-1 correspondence relations that are transitive etc. But that's analogous to Descartes showing that Euclidean geometry an be modelled in logic and arithmetic. As Frege pointed out to Hilbert: the model was not the original. What he failed to notice was that the same objection applies to his logicisation of the ancient cardinal/ordinal (non-fractional) number competence. Anyhow, Piaget had studied those logicians and knew that a grasp of transitivity of 1-1 correspondence was required. Unfortunately he mistakenly labelled it 'conservation' by analogy with conservation of volume, or mass, or length, which was a serious error, since the second case was a meta-property of a continuously varying property, whereas 1-1 correspondence (except in very advanced mathematics not relevant to finite cardinal or ordinal numbers) is restricted to discrete items. Piaget's research (as I understand it) indicated that children did not grasp that 1-1 correspondence is a transitive relationship until they were five or six years old (though not everyone interprets his experimental results that way). May main point is that I don't think there is any evidence that any non-human animal understands the general fact that 1-1 correspondence is transitive. I don't think any *current* robot does either though they may have 'compiled in' implementations of their designers' understanding! Also I don't think any neuroscientist that I have encountered can explain what needs to change in a brain in order to produce an understanding of the transitivity -- not just as an empirical generalisation from examples, but as a necessary feature of 1-1 mappings. This leads me to suspect that during the first 5-6/7 years typical human brains use forms of information processing that transform themselves so as to grasp that transitivity, and other features of 1-1 mappings. But nothing I have heard can explain how brains do that. (I think the Canadian psychologist Lance Rips has made similar points, but he seemed to conclude, wrongly, that for humans that transitivity was somehow innate knowledge.)