I really don't understand this kind of thinking. It ALWAYS comes up on threads like this, and often I wonder if it's men who are saying this or women who themselves are cheaters.
It's cruel to not tell the wife, because cheating is domestic abuse at pretty much every level. It's PHYSICAL abuse because it exposes the wife to STDs, because most male cheaters don't use condoms. It also exposes the wife to the flora from the AP, which can lead to devastating continual bacterial vaginoses for YEARS - that then clear up after they stop having sex with the cheater. Some cheated-on wives get chlamydia that causes infertility, or HPV, which goes on to give them cervical cancer years after they got rid of the cheater.
Infidelity is also SEXUAL abuse. The wife is having sex with the cheater that she would NOT consent to if she knew the cheater was having sex with someone else. It has been proposed that infidelity should be called rape.
Infidelity is also often FINANCIAL abuse. The cheater spends family money on APs, money that could go to paying off the mortgage, or building the kids' college fund. Meanwhile, the wife is often scrimping and saving and going without. Sometimes an affair child arises, which means that part of the money that would ordinarily be going to the wife's children will have to be funneled towards that child. The wife has not consented to any of this embezzlement.
Infidelity is also major MENTAL and EMOTIONAL abuse. Often, wives know subconsciously that something is afoot, and the deep-seated anxiety it engenders wears on their bodies: they start grinding their teeth, they get unexplained headaches, they have chronic gastrointestinal complaints. All of which eventually disappear when the wife leaves the cheater.
There are heavy consequences to being cheated on and those consequences will keep on going for YEARS if the cheater is not stopped.
I can understand that OP might not want to tell the wife (the foul prick might try to bully her) but arguing here that OP SHOULD not tell the wife because it'll harm the wife is ludicrous: HE, the cheater, is literally harming his wife at this very minute! It's also patronising, because you're taking away information that could help the wife protect herself and HER CHILDREN. Most women say they want to know. At least then, they can take action to prevent further damage.