MOTION MOUNTAIN

264 8 thought and language
Challenge 261 d
Challenge 262 s
as everybody thinks that they have the share that they deserve, and it is fullysatisfying,
as everybody has the feeling that they have at least as much as the other. What rule is
needed for three people? And for four?
Apart from defining sets, every child and every brain creates links between the different
aspectsof experience.For example, whenit hears a voice, it automatically makes
the connection that a human is present. In formal language, connections of this type are
calledrelations. Relations connect and differentiate elements along other lines than sets:
the two form a complementing couple. Defining a set unifies many objects and at the
same time divides them into two: those belonging to the set and those that do not;defininga(binary)relationunifieselementstwobytwoanddivides
them into many, namely
into the many couples it defines.
Sets and relations are closely interrelated concepts. Indeed, one can define (mathematical)
relations with the help of sets. A (binary) relation between two setsXandYis a
subsetoftheproductset,wheretheproductsetorCartesianproductX×Yisthesetofall
orderedpairs(x,y) withx∈X andy∈Y. An ordered pair(x,y) can easily be defined
with the help of sets. Can you find out how? For example, in the case of the relation ‘is
wifeof’,thesetXisthesetofallwomenandthesetYthatofallmen;therelationisgiven
by the list all the appropriate ordered pairs, which is much smaller than the product set,
i.e., the set of all possible woman–man combinations.
It should be noted that the definition of relation just given is not really complete, since
every construction of the concept ‘set’ already contains certain relations, such as the relation
‘iselementof.’Itdoesnotseemtobepossibletoreduceeitheroneoftheconcepts
‘set’ or ‘relation’ completely to the other one. This situation is reflected in the physical
cases of sets and relations, such as space (as a set of points) and distance, which also
seem impossible to separate completely from each other. In other words, even though
mathematics does not pertain to nature, its two basic concepts, sets and relations, are
taken from nature. In addition, the two concepts, like those ofspace-timeandparticles,
areeachdefinedwiththeother.
Infinity – and its properties
Mathematicians soon discovered that the concept of ‘set’ is only useful if one can also
call collections such as{0,1,2,3...}, i.e., of the number0and all its successors, a ‘set’. To
achieve this, one property in the Zermelo–Fraenkel list defining the term ‘set’ explicitly
specifies that this infinite collection can be called a set. (In fact, also the axiom of replacementstatesthatsetsmaybeinfinite.)Infinityisthusputintomathematicsandaddedto
thetools of our thinking right at the very beginning, in the definition of the term ‘set’.
When describing nature, with or without mathematics, we should never forget this fact.
A few additional points about infinity should be of general knowledge to any expert on
motion.
A set is infinite if there is a function from it into itself that is injective (i.e., different
elements map to different results) but not onto (i.e., some elements do not appear as
images of the map); e.g. the mapn→2n shows that the set of integers is infinite. Infinity
also can be checked in another way: a set is infinite if it remains so also after removing
one element, even repeatedly. We just need to remember that the empty set isfinite.
Onlysets can be infinite. And sets have parts, namely their elements. When a thing or
Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–November 2015 free pdf file available at www.motionmountain.net