Humans perceive the world as a collection of things that are interconnected in a regulated manner. Due to these rules, our sensory organs can assign properties to individual things or specific combinations of things, allowing for a representation in our thoughts.
As an abstract model of this situation, we introduce the concepts of object and condition, where an object \(x\) can fulfill a condition \(C\). In this case, we call \(C\) a property of \(x\) and write \(x:C\) for shorthand.
While objects and conditions initially appear to be clearly distinct, a brief consideration shows that the boundaries are not sharp. In fact, a condition \(C\) may satisfy the condition “\(C\) is a property of \(x\)” which flips the original roles: now a fixed object \(x\) detemines a property which an arbitrary condition may possess. Conditions thus turn into objects of other conditions. Since the same construction can be applied again, conditions on conditions also take on object characteristics, and so on.
A common everyday example can be found in job descriptions in a newspaper, i.e., lists of properties that can be fulfilled by certain people. A job advertisement can thus be understood as a condition on people. However, when a job seeker reads the job section of a newspaper, they are only interested in job advertisements that apply to them. This creates a condition on the job descriptions, giving them object characteristics.
The names object and condition, therefore, are not absolute in their usage. In the following, we use the name meta-object for something that can simultaneously have the characteristics of an object and a condition.
Let us proceed with the following thought experiment: assume there is a universe \(\mathcal M\) of meta-objects in which each condition that could be posed on members of \(\mathcal M\) is again a member of \(\mathcal M\). To denote a condition, we use the syntax \(\{x\,\,\text{with}\,\, C\}\) where \(C\) is a condition which may be formulated with the parameter \(x\). The condition is fulfilled for a particular member of \(\mathcal M\) if that member fulfills \(C\) after substituting \(x\).
In a first step, we notice that this universe cannot be empty. In fact, the condition \(U:=\{x\,\,\text{with}\,\, (x:x)\vee\neg(x:x)\}\) denotes a member of \(\mathcal M\) and therefore inhabitates \(\mathcal M\).
Using classical logic, we see that \(U\) is satisfied by all members of \(\mathcal M\) because the condition \(x:x\) is either satisfied or not.
On the other extreme, \(E:=\{x\,\,\text{with}\,\, (x:x)\wedge\neg(x:x)\}\) is satisfid by none of the members because the condition is contradictory.
A much more interesting example pesented by Bertrand Russell shows that the universe \(\mathcal M\) is a ticking time bomb. Russell’s tricky condition, which we will henceforth refer to as \(R\), describes the property of conditions not fulfilling themselves, i.e. \(R=\{x\,\,\text{with}\,\,\neg(x:x)\}\). Since \(R\) is a member of \(\mathcal M\) it either fulfills the condition \(R\) or it does not. In the first case, \(R\) would be a self-fulfiller, thus violating condition \(R\), making \(R\) not a self-fulfiller — a contradiction. In the second case, \(R\) is not a self-fulfiller, thus fulfilling condition \(R\), hence is indeed a self-fulfiller — an irresolvable contradiction.
Russell’s example thus shows that to maintain logical consistency, we should not completely forget the distinction between objects and conditions at least to the extent that self-referencing conditions are suppressed. In fact, when formulating a condition, one typically thinks of an already existing collection of objects to be tested and does not consider that the completed condition will itself belong to the test objects.
In \(\small\mathbb M\)ATh, we adopt this idea in two ways: First of all, we require that every condition needs to specify which objects it addresses before itself can be formulated. This changes the notation to \(\{x:A\,\,\text{with}\,\,C\}\) where \(A\) is the a-priori restriction on the potential fulfillers of the condition .
Secondly, we confer the status “element” to those entities which may fulfill conditions. All natural numbers belong to this class and also many functions and conditions are elements, though not all of them. Prominent exceptions are the condition to be a relation or the condition to be a certain algebraic structure (like a group, a ring or a vector space).
Finally, we identify conditions and sets in \(\small\mathbb M\)ATh which explains the use of the set notation for conditions. In particular, we can translate \(x:C\) either as “\(x\) fulfills \(C\)” or as “\(x\) is in \(C\)” depending on our perspective on \(C\).
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