From my notes:
משניות in עירובין – The תנאים were careful to make sure that there were parts of their city where there was no עירוב so it would be אסור to carry in order to make sure that people would not forget that there is an אסור to carry on שבת.
מרדכי – Every רב and every Jewish leader should make sure there is an עירוב in his city.
תשובות רא"ש כלל כא סימנין ח,ט – Was so strongly for building an עירוב that he threatened to put a רב in a nearby city into חירם if he didn't, and said that if the רב wouldn't do it he would be חייב מיתה since he was going against the חז"ל who say there's no problem with making an עבד.
Its אסור to carry from a (רה"ר (רשות הרבים to a (רה"י (רשות היחיד, the opposite, and within a רה"ר.
Its מותר to carry within a רה"י, from a רה"י to a (מ"פ (מקום פטור and the opposite, and from a רה"ר to a מ"פ and the opposite.
The רבנן split a מ"פ into two domains;
- one is still considered a מקום פטור
- the other is a כרמלית which is basically like a רה"ר
שלומו המלך ובית דינו - Made a גזירה that although מן התורה you can carry from a רה"י to another רה"י, they made it אסור דרבנן because people started to think that you can carry from any רשות to another רשות. However, made an עירוב חצירות within an enclosure in order to combine the רשות היחידים through food (usually people use מצות since they last a long time, and are זכין לאדם שלו בפניו so every one owns it).
If the whole point of the שלומו's גזירה is to stop people from not knowing about carrying, isn't there a problem of the עירוב חצירות since most people don't even know it exists? Yes, we should try and educate people.
Enclosure for an עירוב
רה"ר מן התורה – Two reasons why its hard to make an עירוב;
1) Needs to be a wall of at least 10 טפחים high.
2) (אתו רבים ומבטלי מחיצות כשיש פרצה של יתר מעשר (אמות – When there is a hole or breach in the enclosure wall which is greater than 10 אמות ?wide, if it is a place where people go through the wall, that side of the wall is halachically considered non-existent, and the entire enclosure isn't good.
רה"י – Nowadays an enclosure for a רה"י is generally created using a צורת הפתח, which is a combination of two posts with either a wooden bar or a string running across the top of both (they need to be above the posts), connecting them.
Why would a צורת הפתח not be a viable option for a רה"ר מן התורה? First we need to understand the מקור for a צורת הפתח.
:עירובין ו:, יא.,יא – A צורת הפתח can work as a dividing wall and an enclosure.
שו"ע אורח חיים שסב – You can even use string on top.
How does it work as an enclosure, isn't almost completely open?
ראשונים, אחרונים – We view it halachically as an enclosure, since if you open the door of your house your house is still a רשות היחיד despite the fact that its almost completely open.
.עירובין יא – A צורת הפתח can definitely work as a מחיצה המפסקת (dividing wall) מן התורה. We learn this from כלאים, since if you have two types of plants which would be considered כלאים with nothing between them, you can build a צורת הפתח in between them מן התורה and they aren't considered כלאים, therefore we apply the same idea to עירובין.
Therefore, מן התורה, if you have an enclosure of three walls (eg a מבוי, a courtyard surrounded with walls), מן התורה you can use a צורת הפתח as a third wall, and if you have two walls you can use two צורת הפתחs as two walls.
חזון איש – Based on 'תוס on .עירובין ו says that you can have a full enclosure of צורת הפתחים on 4 sides.
Most רשאונים, אחרונים – No, since its based off of כלאים, and by כלאים its not an enclosure, its just a מחיצה המפסקת.
פרי מגדים, משנב ברורה, etc. - Therefore a צורת הפתח can't divide a רה"ר, and you need a wall as an enclosure for a רה"ר.
ר"י עירובין יא., רא"ש, רשב"א, ר"ן - Can be used for a רה"ר דארייתא.
חזון איש סימן ע אות יג – Agrees.
שו"ע הרב אורח חיים שסד – Is also מיקל.
ר האי גאון, רמב"ם, ריצב"א, רבינו חננאל – Can't be used.
באור הלכה שסב:י ד"ה כשכל – Discusses this 'מח, says it is possible that they agree they can be used for רה"ר although probably not, so says its a ספק.
פרי מגדים שסג – Also says its a ספק so we are מחמר since its דארייתא.
What about if there are walls around the community or you hold that צורת הפתח are ok דארייתא?
אתו רבים ומבטלי מחיצות
.עירובין כ.,כב – According to ר' יהודה and ר' יוחנן its פסול דארייתא, according to חכמים and ר' אלעיזר its only a פסול דרבנן.
באור הלכה שסד – Majority hold its a פסול דארייתא.
רמב"ם יז:י – Its only דרבנן.
What about if you close the doors at night of the city so אתו רבים doesn't apply at night?
שו"ע שסד:ב - It solves the problem since it needs to be constant.
Some say even if you could close the gates and you don't there is no problem.
To solve this problem nowadays some Jews go one day a year to set roads and put a sheet across to show that it could be closed, although this is not so פשוט.
רמ"א quoting מהר"ם מרוטנברג – The whole פסול of אתו רבים only applies when the walls that its removing are actually required מן התורה.
Therefore, in a כרמלית since its are מקום פטור מן התורה it doesn't apply.
5 conditions that everyone agrees to that makes a רשות a רה"ר דארייתא;
1) .שבת צט - The main street(s) need to be 16 אמות wide.
2) It can not have a roof (therefore, for eg, if a store on the main street has a canopy over the front then under the canopy isn't included in the width of the street).
3) It has to have less than 3 walls.
4) It has to be owned by the public.
5) :עירובין נט - It has to be מפולש משער לשער (open ended).
R' Moshe Feinstein – Side streets are also called רה"ר מן התורה if the main streets are since it 'spills over'.
Therefore it seems that almost all cities are רה"ר מן התורה.
חזון איש – Buildings can count as walls, so they can be used to solve a רה"ר מן התורה (either because its like three walls or other buildings block it).
משכנות יעקב – Disagrees and says that a building can't be used as a wall, either its a building or a wall.
Most communities aren't נוהג like the חזון איש.
רמב"ן – On #5 You need to have מפולש משער לשער בלי שום עיקום כלל (they need to be perfectly straight all the way through).
Therefore it would be extremely hard to get a רה"ר מן התורה since most cities have streets that curve.
R' Moshe Feinstein – Rejects the רמב"ן since he is a שיטת יחיד.
'רש"י עירובין ו., תוס, many other ראשונים – It needs 600 thousand people to be a רה"ר (they learn this from the fact that there were 600, 000 Jews in the מדבר, where we learn out the idea of carrying from the משכן).
Does this mean only living there or even traversing?
R' Moshe Feinstein, מ"ב – Even 600, 000 living there.
משכנות יעקב – When is there a case of a city 600, 000 people? Very few cities, so would חז"ל uproot carrying שופר on שבת for one or two cities? It must be people living in it.
Therefore, most cities aren't actually a רה"ר מן התורה, rather a כרמלית according to these ראשונים.
רמב"ם הל' שבת, רי"ף – Never mention it.
בית יוסף שמה – If רמב"ם doesn't mention it, its not included.
רמב"ן, ר"ת, רשב"ם, many others – All are מחמר and reject the היתר.
שו"ע שמה:ז, שו"ת אבקת רוכל ס' כט – Rejects the היתר.
שו"ע שג:יח – There is no רה"ר מן התורה nowadays.
How can he say this, isn't 600,000 not a good היתר?
מגן אברהם שמה, ט"ז – The שו"ע is מיקל and holds it is a good היתר.
מ"ב, many others – No, he meant in certain cases that if its a question if its an act of לבישה then in those cases we can be מיקל, but really its not ok.
R' Ovadiah Yosef – Therefore, generally ספרדים shouldn't rely on עירובין.
בית אפרים – The שו"ע actually didn't take any opinion. (This opinion doesn't seem right.)
רמ"א שמו:ג – All our domains today are a כרמלית, and there is no רה"ר דארייתא.
ט"ז שמה ס"ק ו – Requires 600,000 to be a רה"ר מדארייתא based on this רמ"א.
מגן אברהם שם ס"ק ו – Understands the רמ"א this way, and says רוב פוסקים are מיקל.
בית מאיר – The רמ"א is לאו דווקה. (R' Schreiber – Doesn't seem correct.)
מ"ב שמה ס"ק ג, באור הלכה – Don't quote the רמ"א.
שו"ע הרב, חיי אדם, ערוך השולחן, מ"ב, באור הלכה – All agree that the מנהג of אשכנז is that you need 600,000.
ערוך השולחן שמה:יח – Its like a בת קול came out and told us to be מיקל like this its so entrenched in אשכנז.
מהרש"ל יד של שלמה ביצה ג:ח – Was מחמר, proving it from the fact that חז"ל were גוזר you can't blow שופר on שבת on רה"ר, since why would they be גוזר if there are so few places that are רה"ר?
משכנות יעקב או"ח קט:קיט-קכב – Was very against this היתר, and said that if all these גדולים knew about all the ראשונים against it (since they weren't published in their times), they wouldn't have been מיקל. Also, it says נחמיה יג:יד-לא that נחמיה he was angry that ב"י carried on שבת, even though they were only 400,000, and therefore he wouldn't be so angry for only דרבנן, therefore it must have been a רה"ר דארייתא and the היתר of 600,000 doesn't exist. (R' Schreiber – Second ראיה doesn't seem correct, why wouldn't he be angry for a דרבנן?!)
מ"ב שסד:ב באור הלכה ד"ה מאחר שעשה – We assume three things להלכה; that צורת הפתח works in a רה"ר דארייתא, that אתו רבים is only a פסול דרבנן like the רמב"ם, and since the issue is an אסור דרבנן since צה"פ works and אתו is only דרבנן, we are מיקל in the ספק of the היתר of 600,00 and say that without that many people its not a רה"ר.
Rav Shechter, others – What the מ"ב meant was that it was a ספיק ספיקא לקולה. The first ספק is whether a צה"פ works in a רה"ר דארייתא, and the second ספק is even if we hold צה"פ doesn't work in רה"ר דארייתא, maybe we do need 600,000 to make a רה"ר דארייתא, and we hold like the רמב"ם that אתו רבים is דרבנן (?end).
ערוך השולחן שמה:כ-כב – Says that because the עירובין have spread throughout all the communities for many years its like a בת קול has come out and said its ok, and if פוסקים tried to go against it, the people would say they are crazy and its a מצוה to be מלמד זכות for ב"י, so therefore he comes up with this קולא; that even cities that have 600,000 aren't רה"ר since the city needs to have only two major streets that cross each other perpendicularly, and the rest around it need to be minor streets.
- Also, a רה"ר needs to have one central place where everyone goes to buy food etc (like a market), so nowadays since its spread out they aren't רה"ר.
What about a city? How do you rent the streets?
גמרא – Go to the governor (מושל העיר).
What about the problem of their homes being inside the enclosure (since the צה"פ circles the entire community), and the מושל העיר doesn't own everything?
We actually buy the entire city from the מושל העיר, since he really owns everything.
What about in a democracy?
R' Moshe Feinstein – In a democracy its even better, since the government rents out everyone's homes, so we buy the entire city for שבת.
YU Museum exhibit on Eruvim: