Re the New Scientist article 'it doesn't add up', (New Scientist, Vol 207, No. 2773, pp34-38) is fascinating. The whole area of infinities and number theory is wonderful and, as a science trained artist, has captured my imagination since childhood.
The Nature of the Infinity Boundary.
I would agree with Doron Zeilberger - in the sense that its a nonsense to consider that as we count up numbers we eventually continue until 'infinity'. Of course, there is no magnitude limit!. We can always add one to whatever an 'infinite' number is. So what is the nature of the boundary of this infinity?
Let us consider that, in the extremely simple case of integer number, the infinity boundary in this case may be understood to describe a 'realm of quality' - a dimension-like entity with an 'attribute boundary'.
We can make sense of this boundary, which defines the reality of infinity (of integer number), if it describes the boundary of an unambiguous 'realm of all integer number' - an attribute boundary describing a realm of quality of number - in this case the realm of 'All Integer Number'. The set of all integer number if you like. But in this case the 'set' is not merely a collection of numbers, it is invested with dimension-like properties.
So the reality of the 'infinity', in the case of integer number, can be understood by observing it from the 'ouside', in which case we see a dimension-like attribute boundary, a phase boundary, rather than any kind of simplistic, ambiguous and nonsensical 'magnitude limit' (which has no limit), which is of course, nonsense. As an integer number becomes increasing large it becomes more dimension-like (cf limit condition as the dimension-like realm of quality of All Integer Number), whereas 'smaller' numbers are more 'particle' like with the quantum of 1 as the smallest integer unit.
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