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Mon Jan 2 07:34:30 CET 2012
r of irrationals that can be represented at all no matter how. For a certai=
n ontology useful to constructivism, it can be said that the uncountably ma=
ny irrationals that are inexpressible also don't exist.<br>
<br>
This leaves you with countably many rational numbers and countably many irr=
ationals, that can't be represented in a finite format.<div class=3D"im=
"><br>
<br>
<blockquote class=3D"gmail_quote" style=3D"margin:0 0 0 .8ex;border-left:1p=
x #ccc solid;padding-left:1ex">
We could also debate over whether infinity is exactly represented.<br>
When some math operation overflows (exceeds the range of floats), the<br>
result assigned is inf.<br>
</blockquote>
<br></div>
Every float represents a range of numbers. The difference with infinities i=
s that they represent half-intervals, that is, a line bounded only on one s=
ide.<div class=3D"im"><br>
<br>
<blockquote class=3D"gmail_quote" style=3D"margin:0 0 0 .8ex;border-left:1p=
x #ccc solid;padding-left:1ex">
That's not the definition of infinity either: Take the set of real numb=
ers R and the ordering operation <, then add an additional point "i=
nfinity" such that for any x belonging to R, x < infinity.<br>
</blockquote>
<br></div>
You should know that there are several competing definitions of infinity fo=
r real numbers (not considering other number systems in which this definiti=
on doesn't work).<br>
<br>
There are three definitions of Real numbers (R) in common use=A0: one witho=
ut any infinite number, one with two infinite numbers as endpoints, and one=
with a single infinite number without a sign. There are different motivati=
ons for the use of each of those three sets. There's no definition that=
fits all purposes, though the one without infinite numbers at all is consi=
dered generally =ABcleaner=BB in the field of pure math.<div class=3D"im">
<br>
<br>
<blockquote class=3D"gmail_quote" style=3D"margin:0 0 0 .8ex;border-left:1p=
x #ccc solid;padding-left:1ex">
So, the inf in the float definition only represents "infinity" de=
fined relative to the finitely countable set of numbers that can be represe=
nted as floats<br>
</blockquote>
<br></div>
Yes, except NaN.<br>
<br>
You'll also find out that certain definitions of infinity that applies =
to the whole set of Reals also are relative to just that set, and don't=
work as-is for all possible extensions of Reals=A0; for example, Complex n=
umbers don't have a single coherent definition of less-than and greater=
-than anymore, because all you can do is extract features of Complex number=
s and compare those features as Reals... thus you need more specific defini=
tions (and there are more possibilities of them).<div class=3D"im">
<br>
<br>
<blockquote class=3D"gmail_quote" style=3D"margin:0 0 0 .8ex;border-left:1p=
x #ccc solid;padding-left:1ex">
not the actual infinity as represented in your head :)<br>
</blockquote>
<br></div>
How do you know what's in people's heads=A0?<div class=3D"HOEnZb"><=
div class=3D"h5"><br>
<br>
=A0______________________________<u></u>______________________________<u></=
u>__________<br>
| Mathieu BOUCHARD ----- t=E9l=E9phone=A0: <a href=3D"tel:%2B1.514.383.3801=
" value=3D"+15143833801" target=3D"_blank">+1.514.383.3801</a> ----- Montr=
=E9al, QC</div></div><br>_______________________________________________<br=
>
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<br></blockquote></div><br>
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