I have through the assistance of those that took the time and effort to edit and critique my article concerning oats became aware that I had allowed myself to utilize language and reasoning which, though readily understandable to me, is of the more foreign type of logic to those who don't inhabit my own head, that is, everyone else. As such, it behooves me to explain what exactly I am thinking. I should note that it would be sheer arrogance to present this whole way of thinking as a chiddush, thus, it seems to me that I have merely applied a new lashon to what already exists. However, since this may help someone in their learning, it is mandatory to say even this flawed imitation, for it has helped me. I shall in this article give a brief summation of the application of set theory within shas along with several examples to prove its practicality.
Set theory is a branch of mathematics which deals with collections of objects. The general thesis of the application of set theory to Talmudic analysis is, therefore, that halachah is simply a collection of cases. Each set is mapped to either a deed or the negation of a deed. That is, in this case do this, in this case don't do this. For example, if an item is in the set of “not mine,” then I may not bother or take it. Therefore, halachah and Talmudic analysis is merely the act of mapping the cases into its proper set, with the Mishnah being concerned as to the definition of these sets, while the Gemara is more concerned as to clarifying the precise sets that are defined within the Mishnah.
In order to apply Talmudic set theory, we must first define how to construct a set. There are two ways to build sets. The first way is called the roster form which gives all the members of a set. The second method is through builder notation which gives the exact definition of the set through set notation. In order to prove the proposition, we must be able to see these theories within halachah. As an example, in hilchos Shabbos we are given forty minus one melachos. This is the roster form where we are given each member of the set, which is in actuality a family of sets, for if an activity is not a case which belongs to one of the melachos, then it is not dorisa forbidden on Shabbos. Likewise, when the Talmud lists the times where the Mishnah states pattur and puttur means muttar, this is in the form of a roster form with every element of the set being listed.
In addition, we can see within the Talmud the application of the use of set building in order to explain the elements within a set. For example, we see in brochos the following times which are built with set builder notation. All views are based on the first Mishnah and represent the times when one can say kry shema.
R. Eliezer = {x:Tzos HaCocovim < x < End of the first watch}
Rabbonin = {x:Tzos HaCocovim < x < Chatzos}
Rabban Gamliel = {x:Tzos HaCocovim < x < Dawn}
The Mishnah then comes and clarifies that the Rabbonin agree with Rabban Gamliel. They merely imposed a stringency whereupon we have another set defined. This set is defined through set builder notation as well and may be expressed as thus.
Drabbanas on time = {x:x = chatzos then the dorisa is dawn}.
To briefly translate, the set of drabbanas on time is defined as any time which is chatzos. The dorisa time for the performance of the mitzvah is dawn, and the original time is a drabbana as drabbanas on time, as we have named this set, is a member of the set of gezeros as that set is defined as such.
Gezeros = {x:x is a stringency upon a dorisa to protect it}
We could also express this as the f(x) = if x is dawn, then y is chatzos and y is in the set of drabbana.
Additionally, we could have functions that map chatzos unto its sets as Rabban Gamliel is the halachah, chatzos is the lchatchila time thus belonging in the set of lechatchila conditions, and dawn is moved from the lechatchila to the bedieved time.
At the end of the Mishnah it asks what is the reason that we would establish this rule. The reason being that the Rabbonin wanted to keep us far from sin. There are largely three drabbanas, two of which is passed by the chachumim and one of which is not. The relevant one here is the gezerah, and this statement is again highly understandable from the point of view of set theory. As we remarked earlier without explanation, these are gezeros, and Rabban Gamliel's statement serves to assign the drabbanas on time to the set of gezeros, making drabbanas on time a subset of gezeros. This naturally means that when there is a statement on a drabbana gezerah that this applies to all the drabbanas on time.
For example, in the case of human dignity, gezeros are suspended. A very nogeah example of this is tearing toilet paper on Shabbos. Because the feces are on oneself, this is in the set of human dignity. We have the function f(x) = if x is in the set of human dignity then gezeros are suspended. Seeing as how feces on oneself is indeed in the set of human dignity, the rabbinical gezeros are therefore suspended. This permits the tearing of toilet paper off the line in an unusual manner in order to wipe oneself, which can be derived purely through set theory and formal logic.
This of course leads to an answer on how halachos can be lost and then recovered such as after the death of Moshe Rabbeinu where 3,000 halachos were lost whereupon they were rediscovered. The halachos are, for the most part, derived based upon logic from the mikra and halachah lmoshe msinai. If one had a perfect knowledge of the mikra, Talmudic logic, and the halachah lmoshe msinai, it is evident from this midrash that one could resurrect the entirety of halachah merely from tradition and logic alone.
We can also utilize set theory in the laws of acquisition in order to explain a few concepts. In the first daf of Bava Metzia, which I, if I can find my old notes, like to show the transcribing and then the explanation of what I transcribed of the first daf or so into mathematical logic notation, which I believe is useful for comparison's sake as well as the very act being useful for myself in appreciating and understanding the logic of the Gemara. This Mishnah is eminently useful in that it displays several other rules that I would like to explain in brief as it relates to Talmudic logic.
The Mishnah is here very seemingly repetitive and redundant saying both I have found it and all of it is mine. Not only that, but our Mishnah decides to repeat itself twice. The Talmud here presents what I find to be the beautiful subtle illustration of a rule not only of Mishnayos but of the school of logic of Rabbi Akiva may he be remembered for good. The initial assumption by the casual reader is that our Mishnah is merely saying what each of the disputants says in the first case. When it says, “this one says,” the one mentioned is the disputant just as in the later case of uktzem btallis. Indeed, our Mishnah first states that these are one case. There is a union between the sets of uktzem btallis which as we see from the Rosh, is presumed to be the case of a tallis found in a non-Jewish city. While other Mishnayos, such as the ones in Shabbos, the repetition is much more blatant. Here the repetition would not arouse the curiosity of any but a true Talmudic scholar, as it should. The Talmud will then exposit later that these are actually two cases. In the set of uktzem btallis, there are two subsets: that of finding and that of buying and selling. We can, therefore, find a general rule. The repetition of any statement serves to bring a new case. This is so that the Mishnah, despite being written down, is in an encryptic language and thus requires tradition and reasoning derived from tradition to interpret.
We can, therefore, apply this principle in a general sense to if a set is expressed via a roster form rather than a set building form. When a set is expressed in a roster form, this is for three reasons. The first of all, is that each of the expressed members of the set is singled out in some way. The best example of this is hilchos Shabbos where the mikrah applies the set building form and bans all melachah, which we derive through a gezerah shvah which means the activities involved in building the Mishkan. The merciful one then comes and singles out kindling by which we can see a rather partial version of the roster form.
The reasoning for this is multifold. First of all, by expressing in the set building form, the mikra is kept not only short, but dependent upon the Torah Shebaalpeh as the full rule isn't even given, only hinted. This keeps the mind sharp and ensures the Torah cannot be stolen. Second of all, this is the reason why the Mishnah expressess the melachos as forty minus one. It is to teach us two things. First of all, because it is singled out in the roster form, that which applies to kindling applies to every melachah, namely, that each and every av melachah is worthy of its own chatas. This is a major rule as often members of sets combine to prohibit or permit. See for example terumah, challah, bikkurim, and terumas maaser. Because these are generalized by the mikra as terumah they make a set, and indeed it seems to me from the Rambam Teromos 10:4 that one is also liable for the fifth.
This is alluded to in the daf of Bava Metzia that we were just discussing. For, it questions the need of both statements, both I have found it, and all of it is mine. To be brief, these are necessary to prevent seeing from being a kinyan. I have elaborated on the issues with seeing as a kinyan elsewhere in my unpublished writings.
This introduces two logical principles. First of all, every member of a roster constructed set must bring something new to the table. Where space in both print and memory is finite and cases are infinite, each member of a set serves to define the rule of the set. For example, in hilchos Shabbos, each of the listed av melachos has at minimum thirty-nine tolodos which are inferred from the av melachah. As the tolodos are defined through rule alone, sifting being any creative intentional labor where one uses a tool to separate larger desired or undesired material from smaller desired or undesired material using gravity, they in some way combine. In this case, they combine in chatas liability.
See also, in our Mishnah where it discusses the case of the donkey. Does it enter your mind that a horse is a different case? Rather, because one could easily infer from here any animal that is either ridden or lead is the same case, the set is defined through its least number of members. We thus find another principle of Talmudic set logic. Thus we have our second principle, a roster set is defined through as few members as possible. This means that if we are given a case with two members, then we can presume this is a set which is defined through roster principles and thus we apply said principles, such as a lack of combining.
It should be noted that there are exceptions to this rule such as in bameh madliken where the list isf mostly exhaustive. This serves as a proof against me and thus proves this concept is worthy only of an article and requires more thought and study before I could write a whole book.
In conclusion, it is my personal opinion that set theory is indeed a valid and useful framework for the analyzing of halachah, Mishnah, and Talmud in order for one to arrive at the ultimate goal and truth of the Torah. That is, the ability to utilize these principles in practice.
Published 4/27/2025