|From:||William Frank <williamf.frank@xxxxxxxxx>|
|Date:||Thu, 1 Mar 2012 11:50:31 -0500|
Does this summarize the foundational statements from John Sowa and Mathew in this thread? |
terms that begin as primitive may become defined,
(in fact these may in fact change places over time: one system may define T in terms of T', and another, vice versa)
people naturally use second order concepts all the time, and this means that we have to deal with the undecidability of some of the things we want to say, and in fact, people know how to do this.
a basic feature of any defined term is the principle of substitutability: you can always replace the defined term with its definition.
However, just to keep things accurate, this principle applies to recursive functions as well, quite simply, if a little awkwardly, when using higher order (undecidable) languages. Here is how it works, for addition:
There is exactly one function f such that f(x,y) = f(y,x), f(0,x) = x, and for every x, y (successor f(x,y) = f (x, successor(y)))
'+' means that unique function f such that …….
so, every time one encounters the symbol “+”, one replaces that symbol with the definiens: “that unique function f such that …."
thus, the principle of substitutability of defined terms with their definiens is sustained in recursive definitions
this example and its equivalents are easy to find in the literature.
so, I was most surprised to find someone saying:
"It is impossible to define addition in terms of zero and successor."
On Thu, Mar 1, 2012 at 10:16 AM, John F. Sowa <sowa@xxxxxxxxxxx> wrote:
Dear Matthew, Chris, and Nicola,
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