Rigour
 To proceed any further with analysis of the confusions of Gödel
it is necessary to begin to relax rigour.
 By relaxation of
rigour, I mean a degree of distancing from empiric evidence. For example, to start discussing the attributes of ‘unicorns’,
I would be in some manner constrained to act as if unicorns existed in the
real world.
 A peculiarity of the religion widely called mathematics is to call it
‘rigorous’ if one follows the rules decided by ‘mathematicians’! In other words, should a ’theory’ start
from an assumption, such as ‘there are unicorns’, and then at
some point a statement occurred which did not ‘fully’ agree with
the original assumption (that unicorns exist), the statement would be regarded
as an inconsistency (with the original assumptions) and
the line of discussion would be claimed to be not rigorous!
 Thus does mathematics far more resemble a codified superstition than
it resembles ‘science’. Science tends to appeal to evidence,
while mathematicians, like priests, tend to appeal to ‘authority’:
theirs! Note how mathematics thus mirrors an approach to ‘knowledge’ similar
to that of the ontological
argument rather than to the investigation of gravity or the conditions
for heavier than air flight.
 To follow through the game of Gödel, and
all those who imagine that they understand or agree with Gödel,
the first level of rigour reduction that I will indulge
(accept) is the assumption that it is possible to separate
objects ‘completely’.
That is, I will pretend that the ’law’ of the excluded middle
has real meaning. It has not, but I will pretend that it has, just as I
could pretend unicorns were real creatures. Many mathematicians unconsciously
indulge this sort of let’s pretend every day at the office. For further
background, read the
error called‘equality’, the
error called‘zero’or‘not’or‘negative’ and the error called‘infinity’ in Why Aristotelian logic does not work.
 The second level of rigour
reduction that I must indulge is the assumption that any
’two’ of these ‘objects’may be ‘equated’.
This includes the notion that ‘objects’ may be matched or ‘mapped’
‘one’ to ‘one’. This means that, given a group of
turkeys and another group such as a group of pencils or formulae or, for
all I can gather, a group of unicorns, one may line up the turkeys and place
one pencil in front of each turkey until you run out of turkeys. If you
happen to run out of pencils, or even unicorns, at the ‘same’
time, you can be sure that you have the ‘same’ number of turkeys
aspencils.
Advice: if carrying out this experiment, it is helpful if you can persuade
the turkeys to keep still.
Consistency
 Ideas of ‘consistency’ are built upon two nonempiric axioms:
the ‘idea’ of ‘separation’ and the ‘idea’ of ‘equality’.
 These ideas are then applied to what are supposed to be two versions of
consistency: ‘absolute’ consistency and relative consistency.
I shall soon show that ‘both’ forms are, in real terms, relative
‘concepts’.
‘ABSOLUTE’
CONSISTENCY
 Absolute consistency is supposed to exist if two
‘structures’ can be mapped one to one in ‘all’ details.
But, of course, the details mapped are chosen by the individual ‘mathematician’
and the precedence given to any details is selected: the details do not
come from the stars with labels saying, “I am a relevant detail”.
 For example, ‘triangles’ are addressed by ‘their’
vertices and sides; little attention is paid to the ink out of which they
are drawn, or the words used to refer to the vertices or the sides. We are
calmly assured by ‘mathematicians’ that these issues are ‘not
relevant’, or that the triangles are somehow ‘abstract’.
 There is no such thing as an ‘abstract’ triangle any more
than there is a ‘perfect’ circle, the idea of the triangle,
at the very least, resides in the real brain of the mathematician, there
encoded in real molecules and activated by real electrical activity within
the brain.
 If the mathematician can list every combination of sides and vertices
and map each and every ‘one’ them to a bunch of words called
‘axioms’, then the mathematician will pronounce the system ‘absolutely’
consistent.[1] Remember, the mathematician chose the
words, and chose to what they should refer, and decided all this assembly
would sit still while this process was carried out. (See also numbers
or objects.)
 Note that this process is, in fact, relative; it relativises the description
to the triangles.
 A triangle may be described in Mandarin Chinese and in English, and the
two descriptions mapped. The triangle is a way of describing a ‘flat’
surface of a table or a country or a map. All are descriptions of reality. None of them ‘are’ the reality they are designed
to describe. None of the descriptions are the ‘same’. As ever,
if the term ‘same’ is used, it means, “ Right now, I don’t
care about the differences”, and that is all that it means or can
mean. (See also the
error called ‘equality’.)
 The way in which counting is applied is arbitrary choice.
Mapping is therefore also arbitrary.
RELATIVE
CONSISTENCY
 Cantor based his work on a way of
‘counting’ called one to one mapping.
With a little finagling, he ‘proved’ that there were just as
many square numbers as ordinary counting numbers. That is, there are as
many ‘square’ numbers like 1,4,9,16,25 etc as there are numbers
like 1,2,3,4,5 etc; [2] his reasoning being that for each
number such as 10 or 15, you can form a number such as 100 or 225! [diagram
below.] Attend carefully to the dotdotdot at the end of each row, which
is designed to suggest that this process just goes ‘on and on’.
This is nonsense, but it is rather subtle nonsense and it has oft been
repeated since. Hilbert, another of the modern high priests of maths, referring
to this wondrous achievement and seeing the opposition of more downtoearth
workers said, “No one shall drive us from this paradise which Cantor
has created for us”. [3] I intend to do just that.
JUST
HOW MANY NUMBERS ARE THERE?
 Consider ‘a’ plant for manufacturing cars.
There are plans, which are followed in the production process, and there is the factory system that acts upon matter, such as steel and
plastic, to form and assemble the cars.
 With numbers, there also exist a series of rules to be followed,
often termed an algorithm, (see also lists
of instructions) and there is a human who carries
out the process and writes down or ‘thinks’ of the various
real numbers.
 Consider some super mathematician who sets about writing numbers
down, starting at the number one and continuing until they ran out of
chalk or ink. So they go down to the shops for more ink, and then they
run out again. In due course, every factory in the world is devoted
to producing more ink, and ever more writers of numbers are drafted.
‘Ink’ is manufactured from iron filings and cow dung, but
still more numbers can be produced. Space ships are sent out for more
matter.
 Problem: eventually, whether assuming a room
or a planet or a finite universe of resources, you will run out of matter
with which to write numbers. Whatever else, you will get rather tired,
or die, or consider, “Damn this for a game of soldiers, I’d
rather have a meal, or even watch football, than this”. And this
is assuming that you hadn’t already used up the football and the
meal for writing numbers. Thus, the idea of counting
‘for ever’ is not viable in the real world. The common
concept of a ‘completed’ ‘infinity’ is not empirically
viable. (For more detail, see the
error called ‘infinity’, and also the
Logic of ethics, endnote
6).
 We have the nice rules for constructing numbers but, just like the
car manufacture, any particular number still has to be manufactured
out of real matter, taking real effort and real time (see also box
II  relativity)
JUST
WHAT IS THE BIGGEST NUMBER?
 Strangely, we may, with our algorithms,
manufacture numbers beyond all useful meaning. We may produce a number
greater than the number of ‘grains’ of sand on all the shores
of this planet, or even a number that is greater than the number of
‘atoms’ in the known universe. Such an example has been
suggested by a mathematician and named a ‘googol’.[4] It is defined as 10 raised to the power of 100, that is 10^{100}.
This is therefore a number rather like the word ‘unicorn’
and like some uses of zero, in that it does not point at anything in
the real world (further discussion to be found in the
error called ‘zero’...).
 Be clear that we can manufacture arbitrary
numbers, ‘way out there’, far beyond any number we could
ever construct if we start from the number one and continue to count
until we run out of matter.
 Further, because we can produce such a number does not mean that the
number indicates anything real beyond ‘the’ number recorded
upon the paper or an electronic billboard. The number can easily have
a similar status to the word ‘unicorn’.
Countability
and mapping
 To recap: Cantor has defined number in terms
of what may be mapped to what else. (See also paragraphs
106 and 113).
 A typical example is the one referred to above, where Cantor maps
the integers to the square numbers. Cantor appears to misunderstand
that, because he has rules for producing any particular number or ‘square’,
just as the car manufacturer has plans for producing various particular
types of car, it does not follow that there are somehow ‘out there’
the ‘same’ number of square numbers as there are integers.
Every square number, and every integer, must first be constructed in
some real manner out of matter.
 Just which numbers humans choose to construct at any particular time
is a matter of current decision. The numbers are not somehow ‘out
there’ waiting to be picked up like so many stones upon a beach.
 The matter of mapping one number to another is little different to
mapping turkeys to pencils or cars to gloves; but first you must always
catch your turkeys or manufacture your cars.
 Cantor’s mapping thus becomes a mere tautology [5] which amounts to:
If I decide to manufacture the number 2, I can also then manufacture a number
4 (which I will call the square of 2) by applying a particular set of instructions.
This is a piece of information we already had, once ‘square’
had been defined to mean, “Multiplying a number by ‘itself ’” (see
also the error called
‘equality’). It does not follow meaningfully that “there
are the ‘same’ number of square numbers as there are integers”
at this particular moment of time. Without care, it is rather easy to come
to foolish conclusions if you do not map words very carefully to real matter.
 The existential [6] statement that there are the ‘same’ number of squares as there are integers is an empiric
claim that has the potential to be tested by going out and counting the
damn things. If anyone is inclined to carry out this investigation, I am
very willing to bet good money that they will not find the same number of
each type of object. I would make a fair guess that they will find rather
more integers lying around than square numbers. Those who claim that the
numbers are ‘the same’ when used for manufacturing both integers
and squares have clearly confused potential with fact: A parallel, or should
I say a mapping, with the ontological
argument.

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