Pre marriage vows and children I think I "could" of forgiven a ONS but wouldn't of as I think the long term effect of being the "victim" of infidelity would have been too erosive to my feelings of self worth... I deserve far more than that and I don't particularly want a cheat and liar for a partner... I would have moved on, grieved for the partner and future I thought I had, but didn't, and then found someone new!
Now post marriage vows, I feel we have a special bond and it would be broken forever with a ONS or affair..
Now we have children I feel it would be an utter betrayal of our whole life... Although I can't imagine how hard it would be on the children...
So no, I would never forgive and stay
I realise that you are at a difficult time in your life but you need to say goodbye and grieve for the future you thought you had... You will find a better future...
Whilst at the same time I do believe that people can make mistakes and not repeat them, actions have consequences and he will need to learn not to repeat his mistakes with someone new, not you, IMO
??