*Perpetuum mobile* **(***π***0,** *πI***,** *πI I***,** *πIII***)**

*A perpetuum mobile is a hypothetical machine, the movement and efficiency of which would contradict certain recognized laws of physics. The most faithful meaning of the Latin name ("forever moving") perpetuum mobile type 0 (π*0*) would not be stopped in spite of the existence of frictional forces and resistance to motion. A perpetuum mobile type I (πI) would do work from nothing (or almost nothing—efficiency greater than 100%) against the principle of conservation of energy and against the I law of thermodynamics. A perpetuum mobile type II (πI I) would be 100% efficient and would be against the Kelvin principle. On the other hand, the perpetuum mobile type III (πIII) would have efficiency lower than 100%, but higher than the maximum efficiency predicted by the Carnot principle.*

Most often it only stands out *perpetuum mobile π<sup>I</sup>* and *πI I*. However, *perpetuum mobile πIII* is crucial for this article. Sometimes the term *perpetuum mobile* of type III is understood as *π*0. However, the proposed terminology seems clear and consistent with the meaning of the Latin words *perpetuum mobile* for *π*<sup>0</sup> and consistent with the classification of various ways of doing work by *πI*, *πI I*, *πIII*—while machine *π*<sup>0</sup> does not do the work. A similar notation to *perpetuum motion* will also be applied to the *π*↑ process where heat spontaneously flows uphill (towards higher temperature). Such a process, despite its similarities, does not have to be automatically equivalent to *πI I*.

Laws of physics, such as the thermodynamic principles under consideration, are subject to some kind of experimental verification. Unfortunately, from a formal point of view, these principles are not subject to truth proof like mathematical theorems. Nevertheless, the two principles may result from each other. Unfortunately, the implication of principles is something more complex than the material implication of Table 1, that depends only on logical values.


**Table 1.** Table of logical values of material implication and (material) equivalence.

Two principles may be subject to a relationship of formal implication. The formal implication related to a specific theoretical system implemented by certain models becomes a semantic implication. A provable formal implication is called syntactic implication. Kurt Gödel proved the theorem that a semantic implication is a syntactic implication. In other words, a model-based system is needed to prove formal implication. In the case of thermodynamic principles, models are sets of processes (diagrams) that follow a given principle. Thus, the formal implication of the two principles can be written as follows:

$$Z\_1 \mid \to Z\_2 \Leftrightarrow \forall d \in \Omega \; : \; d \in \{Z\_1\} \Rightarrow d \in \{Z\_2\}, \tag{9}$$

where {*Z*} stands for the model, i.e., the set of processes consistent with the *Z* principle. If there are several (or many) models, the formal implication should apply for all *Z*<sup>1</sup> models, and there should be at least one *Z*<sup>2</sup> model containing the same *d* process. The symbol of formal (syntactic) implication |→ used here was created from the combination of the symbol known from the formal logic and the usual symbol of the implication arrow → (see [39]). Strictly speaking, this symbol in LaTeX transcription is a combination of the sign | with the sign → using a negative space ("\med \!\rightarrow"). In formal logic, the

symbol |= is also used for the semantic formal implication. In model theory, this symbol is synonymous, but it is used to constitute a sentence, thesis, or a principle within some system [40]. For example, the fact that in the system of theoretical thermodynamics T the Kelvin law is true we write as T |= *K* or more precisely T |= (*CII* |→ *K*), where we assume the truth of the *CII* principle.

We can see that the system of models (sets of processes) allows us to reduce the formal implications to the material implications. The significant difference, however, is that the material implication must hold true for all the processes (diagrams) of the system to which the formal implication relates. This also applies to transposition law, which is used in the scheme of proof by contradiction:

$$Z\_1 \mid \to Z\_2 \Leftrightarrow \forall d \in \Omega \; : \; d \notin \{Z\_2\} \Rightarrow d \notin \{Z\_1\}. \tag{10}$$

Thus, in order to prove the implication of the principles, it is not enough to point to the *d* process, which contradict the *Z*<sup>2</sup> principle, and to show that it implies a contradiction with the *Z*<sup>1</sup> principle (compare Figure 2a with Figure 2b). It is also worth noting that proof by contradiction is not the most valued method of proof. Constructive proof that follows the original direction of implication is the most valued. There is even an orthodox version of logic (intuitionistic logic), that rejects the law of the excluded middle—including proof by contradiction. There have even been attempts to integrate intuitionism into the mathematical proving system. The Brouwer's program is the best known in this respect.

**Figure 2.** Schematic representation of the relationship between two physical principles *Z*<sup>1</sup> and *Z*<sup>2</sup> in a set of virtual physical processes Ω ({} brackets to distinguish the model from the principle are omitted here): (**a**) The case of implication of principles means the inclusion of appropriate sets (models). (**b**) A case of principles that do not implicate each other. The counterexample in the form of the *d*<sup>1</sup> process disproves the implication *Z*<sup>1</sup> |→ *Z*2, despite pointing to the *d* process that would allegedly prove this implication by contradiction.

So, in order to prove the *Z*<sup>1</sup> |→ *Z*<sup>2</sup> implication of the principles, it is necessary to show that the sets of processes conforming to these principles (their models) are subsets {*Z*1}⊂{*Z*2}. The set of the resulting (often weaker) principle should, therefore, be a set greater than or equal to the set of the previous (often stronger) principle. It is a little easier to rebut the implications of two principles. For this purpose, it is enough to indicate the *d*<sup>1</sup> process, which belongs to the {*Z*1} set, and does not belong to the {*Z*2} set (see Figure 2b).

The principles are formally equivalent when the formal implication acts both ways:

$$Z\_1 \leftrightarrow Z\_2 \Leftrightarrow (Z\_1 \mid \rightarrow Z\_2) \land (Z\_2 \mid \rightarrow Z\_1) \Leftrightarrow \forall d \in \Omega \; : \; (d \in \{Z\_1\} \Leftrightarrow d \in \{Z\_2\}).\tag{11}$$

Usually the equivalence of the three principles is proved by a looped chain of implications *Z*<sup>1</sup> |→ *Z*<sup>2</sup> |→ *Z*<sup>3</sup> |→ *Z*<sup>1</sup> (or just as well in the opposite direction). However, this way of proving is prone to error—it is enough to undermine the weakest link in the chain loop for the equivalence proof to be invalid. A more comprehensive approach is to analyze the entire matrix of implications (Table 2).


**Table 2.** Full *status quo* matrix of the mutual implications of the four principles related to the II law of thermodynamics. Question mark "?" refers to the vague *status quo* of implication, and the annotation "(?)" refers to the implications questioned by the author.

We can see that of the 12 implications, 3 do not have a clear *status quo*. The point is that Carnot's principle is not given as an equivalent formulation of the II law of thermodynamics, but as a consequence of this law. Moreover, we see that as many as 6 other implications are being questioned. The point is that the author supposes that the principles of Clausius I and Kelvin are not equivalent to each other and are not equivalent to the II law of thermodynamics. The main goal of this paper is to formally prove these hypotheses.
