Just a little vent

Hey guys just needing to vent a little bit. The father of our one yr old child lives in a different state, from the moment I was pregnant he gave false promises (I will move to you guys but put no effort in and instead moved further away), he says a lot of stuff that sounds good in the moment but never follows through. He has the money and time to visit his new gf on a diff coast but ignores me when I ask if he can help financially. He lies and says he sent son something in the mail and then when I mention nothing came he goes MIA for months. He calls his son once every few months. But the thing is, he is telling everyone he takes care of the baby while I “do my thing” even tho I am a 24/7 stay at home mom and he came to see him only once. He tells people he bought us a house 🤣 he has only sent $200 In one year. I’ll tell him son is sick with a fever and have to do an emergency visit and it took him a month to reply and didn’t even ask how he was. He uses pics I send to post and make it look like he’s an active present dad. Then when we do talk he “cries” and says how much he misses him, and then turns around to praise himself to me how good of a father he is and no other dad would do what he does (like what? 🤣) And I don’t want to be rude to him but it’s a slap in my face as the person who does EVERYTHING alone. I asked if he could participate in sons cultural activities and he says “ehhh let’s just do new traditions but wants me to do all his cultural activities and just never contributed to what I wanted to do for son for his culture. I had a very inconsistent father and that’s what it’s looking like he will be, someone who is in and out as he feels but tries to take credit to look good. I know it’a little compared to some situations. But I don’t know going forward if I should allow him to continue to be In and out and have empty promises esp once son understands it, or if I should start to set boundaries and stop chasing him to care about his son.